Integrable reductions of the dressing chain

In this paper we construct a family of integrable reductions of the dressing chain, described in its Lotka-Volterra form. For each $k,n\in\mathbb N$ with $n\geqslant 2k+1$ we obtain a Lotka-Volterra system $\hbox{LV}_b(n,k)$ on $\mathbb R^n$ which is a deformation of the Lotka-Volterra system $\hbox{LV}(n,k)$, which is itself an integrable reduction of the $m$-dimensional Bogoyavlenskij-Itoh system $\hbox{LV}(2m+1,m)$, where $m=n-k-1$. We prove that $\hbox{LV}_b(n,k)$ is both Liouville and non-commutative integrable, with rational first integrals which are deformations of the rational integrals of $\hbox{LV}(n,k)$. We also construct a family of discretizations of $\hbox{LV}_b(n,0)$, including its Kahan discretization, and we show that these discretizations are also Liouville and superintegrable.

The dressing chain is an integrable Hamiltonian system, which was constructed in [20] as a fixed point of compositions of Darboux transformations of the Schrödinger operator. It was shown in [7] that after a simple linear transformation it becomes a Lotka-Volterra system, which is a deformation of the Bogoyavlenskij-Itoh system [11,4]. For the integrable reductions of the dressing chain which we will study here, the Lotka-Volterra formulation is the most convenient; also, we will use many results from [18,13,6,7], which are all written in that formulation.
For integers n and k, satisfying n 2k + 1, the Hamiltonian system LV(n, k) has as its phase space R n , which we equip with its natural coordinates x 1 , . . . , x n . It has as Hamiltonian H the sum of these coordinates, H := x 1 + · · · + x n and as Poisson structure a quadratic Poisson structure, with brackets i,j x i x j , 1 i n .
These systems were introduced in [6], where we also established their Liouville and non-commutative integrability (see Definition 4.1), with rational first integrals. For n = 2k + 1 one recovers the Bogoyavlenskij-Itoh system whose deformation, which is the dressing chain, was constructed in [7]. Most importantly, the systems for which n > 2k + 1 can be obtained by reduction from a Bogoyavlenskij-Itoh system LV(2m + 1, m), with m := n − k − 1. The same reduction can be applied to the deformed system LV b (2m + 1, m), leading to a Hamiltonian system, which we will denote by LV b (n, k). The Hamiltonian vector field X H now has the forṁ where all entries b i,j of the skew-symmetric matrix B (n,k) , satisfying |i − j| / ∈ {m, m + 1}, are zero and the other entries are arbitrary parameters. Setting in this system all deformation parameters equal to zero, one recovers LV(n, k). A natural question, studied here, is the integrability of LV b (n, k) for all n and k. For LV b (2k +1, k) the answer is known [20,7]: LV b (2k +1, k) is Liouville integrable with polynomial first integrals which are deformations of the first integrals of LV(2k + 1, k).
The main result of this paper is that LV b (n, k) is on the one hand Liouville integrable, with rational integrals which are deformations of the integrals of LV(n, k), and is on the other hand non-commutative integrable, with such first integrals. See Theorem 3.4 for the case of (n, 0) and Theorem 4.9 for the case of (n, k) with k > 0. In order to establish these results, we need to construct the deformed integrals and show that they have the desired involutivity properties; independence is in fact quite automatic and is proven by a simple deformation argument.
Surprizingly, the construction of the deformed first integrals from the undeformed ones is very simple, and is the same for all integrals of LV(n, k) that we constructed in [6]: from such a first integral F of LV(n, k) we obtain a first integral F b of LV b (n, k) by setting and where the indices of b and x are taken modulo 2m + 1.
The proof that we get in this way first integrals and that they are in involution when the undeformed integrals are in involution needs however extra work, as it does not follow directly from the definition. In the case of LV b (n, 0), studied in Section 3, there is only one deformation parameter β = b 1,n and the action of e D b on the rational integrals of LV(n, 0) which were constructed in [18] can be equivalently described as the pullback of a birational map, which we introduce. Moreover, we show that this map is a Poisson map between the deformed and undeformed systems (Proposition 3.2). This yields the integrability results for LV b (n, 0), since apart from the Hamiltonian, all given first integrals are rational; the fact that these rational integrals are in involution with the Hamiltonian, i.e., are first integrals, can in this case be shown by direct computation (Proposition 3.1).
When k > 0 the above idea can also be used, but some care has to be taken because there are now 2k + 1 deformation parameters, and they can be added one by one, upon decomposing D b = 2k+1 p=1 D (p) , but in order to be able to view at each step the action of e D (p) on the rational integrals, as the pullback by some Poisson map, one has to add the parameters in a very specific order. The reason for this is that in this process the form of the integrals at each step is very important. With this, one gets that the deformed rational integrals of LV b (n, k) are in involution (second part of Theorem 4.2). This system has k + 1 independent polynomial integrals, which are by construction in involution, because they are restrictions to a Poisson submanifold of the involutive integrals of LV(2m + 1, m), but they also have to be shown to be in involution with the rational first integrals. This is again done using the above Poisson maps, but since these do not produce the deformed polynomial integrals, some extra arguments which are again very much dependent on the particular structure of the integrals, are needed. In the end, this proves Theorem 4.9 which says that the deformed systems LV b (n, k) are both Liouville and non-commutatively integrable.
A discrete system associated with the dressing chain was first considered in [2]. As was shown in [2], this discrete system preserves the Poisson structure as well as the integrals of the underlying continuous system. The discretization of the dressing chain was rediscovered in the context of the Painlevé equations in [17], using the Lotka-Volterra formulation of the dressing chain. We will construct in the final section of this paper a class of discretizations of the LV b (n, 0) system, including the Kahan discretization. We prove that the discrete maps which we obtain are birational Poisson maps and that they preserve the Hamiltonian as well as the rational first integrals of the (continuous) LV b (n, 0) system. These discretizations are therefore both Liouville integrable and superintegrable. It would be interesting to study the integrability properties of LV b (n, k), with n > 2k + 1 and k > 0.
The structure of the paper is as follows. We construct in Section 2 the systems LV b (n, k) as (Poisson) reductions of the systems LV b (2m + 1, m), where m := n−k−1, and we show that the inherited Poisson structure Π (n,k) b is a deformation of the Poisson structure Π (n,k) of LV(n, k). In Section 3 we construct rational first integrals of LV b (n, 0) as deformations of the first integrals of LV(n, 0), which were constructed in [18]. We show by using a Poisson map, which we also construct, that half of these first integrals are in involution, establishing both the Liouville and superintegrability of LV b (n, 0). We also give explicit solutions for this system. In Section 4 we treat the more complicated case of k > 0, where we prove again Liouville integrability, and also non-commutative integrability, a notion which is intermediate between Liouville and superintegrability. In this case we use 2k + 1 Poisson maps, which are composed in a very specific order to obtain the results. In Section 5 we construct a family of discrete maps for LV b (n, 0) as compatibility conditions for a linear system, associated to the Lax operator of LV b (n, 0), and show their Liouville and superintegrability. We show that the Kahan discretizations of LV b (n, 0) is a particular case and deduce from this the Liouville and superintegrability of the Kahan discretization.

The Hamiltonian systems LV b (n, k)
In this section, we construct the polynomial Hamiltonian systems LV b (n, k). Recall that n and k stand for two arbitrary integers satisfying n 2k + 1. We construct them as reductions of the deformed Bogoyavlenskij-Itoh systems, which we introduced in [7]; in the notation of the present paper, the latter systems are the systems LV b (2m + 1, m), where m := n − k − 1.
2.1. The deformed Bogoyavlenskij-Itoh systems. We first recall the Bogoyavlenskij-Itoh systems LV(2m + 1, m), which have first been introduced by O. Bogoyavlenskij [3,4] and Y. Itoh [11], and their deformations LV b (2m + 1, m), which we constructed in [7]. In both cases, the phase space of the system is R 2m+1 , which is equipped with its natural coordinates x 1 , . . . , x 2m+1 . Since many formulas are invariant under a cyclic permutation of these coordinates, we view the indices as being taken modulo 2m + 1, i.e., we set x 2m+ℓ+1 = x ℓ for all ℓ ∈ Z. The Poisson structure Π m of LV(2m + 1, m) is constructed from the skew-symmetric Toeplitz matrix 1 A m whose first row is given by It leads to a quadratic Poisson structure Π m on R 2m+1 , upon defining the Poisson brackets As Hamiltonian we take H := x 1 +x 2 +· · ·+x 2m+1 , the sum of all coordinates. The corresponding Hamiltonian system is given bẏ It is called the Bogoyavlenskij-Itoh system (of order m), and is denoted by LV(2m+1, m). Given any real skew-symmetric matrix B m of size 2m+1, define 2) These brackets define a Poisson structure, denoted Π m b , if and only if all uppertriangular entries b i,j := B m i,j of B m , with j − i / ∈ {m, m + 1} are zero (see [7,Prop. 3]). Under this condition on B m , we can consider the Hamiltonian system on R 2m+1 with the same Hamiltonian H and Poisson structure Π m b . It is given bẏ It is called the deformed Bogoyavlenskij-Itoh system (of order m) and is denoted by LV b (2m+1, m). It is clear that setting all parameters b i,j equal to zero, one recovers LV(2m + 1, m). A Lax equation (with spectral parameter) for (2.3) is given by where for 1 i, j 2m + 1 the (i, j)-th entry of the matrices X and M , and of the diagonal matrices ∆ and D, is given by 1 Later on, the matrix A m (and similarly the matrix B m and the Poisson structure Π m ), will have two superscripts; in that notation, A m is written as A (2m+1,m) , and similarly for B m and Π m .
It generalizes Bogoyavlenskij's Lax equation, which can be recovered from it by putting all b i,j equal to zero, i.e., by setting ∆ = 0.
2.2. The reduced systems. The systems LV b (n, k), with n > 2k + 1, are obtained by reduction from LV b (2m + 1, m), where m := n − k − 1 > k. Consider the submanifold N n of R 2m+1 , defined by x n+1 = x n+2 = · · · = x 2m+1 = 0. It is a linear space of dimension n which we identify with R n and on which we take the restrictions of x 1 , x 2 , . . . , x n as coordinates (without changing the notation).
Proposition 2.1. If the entries of the skew-symmetric matrix B satisfy b i,j = 0 whenever n + 1 i 2m + 1, then N n is a Poisson submanifold of R 2m+1 , Π m b . Proof. We need to show that all Hamiltonian vector fields X F := {· , F } m b , where F is an arbitrary function on R 2m+1 , are tangent to N n at all points of N n . This is equivalent to the vanishing of X F [x i ] = {x i , F } m b at all points of N n , for any i with n + 1 i 2m + 1. Since for such i, we have b i,j = 0 for all j, and by the derivation property of the Poisson bracket, which clearly vanishes on the hyperplane x i = 0, and hence on N n .
Since N n ≃ R n is a Poisson submanifold of (R 2m+1 , Π m b ), we can restrict Π m b (as given by (2.2)) to R n , giving a Poisson structure Π (n,k) b , with associated Poisson bracket where A (n,k) and B (n,k) denote the n × n matrices obtained from A m and B m by removing its last 2m + 1 − n rows and columns. Said differently, A (n,k) denotes the skew-symmetric n × n Toeplitz matrix whose first row is given by (0, 1, 1, . . . , 1 Thus, all uppertriangular entries of the skew-symmetric matrix A (n,k) are ±1 and A (n,k) i,j = 1 if and only if n + i > k + j. Also, B (n,k) is the skewsymmetric n × n matrix whose uppertriangular entries b i,j := B (n,k) i,j with j − i / ∈ {m, m + 1} are zero. So when k > 0, the first line of B (n,k) is given by (0, 0, . . . , 0 while for k = 0 it has the form (0, 0, . . . , 0, b 1,n ). We define LV b (n, k) to be the Hamiltonian system with Π (n,k) b as Poisson bracket and H = x 1 +x 2 +· · ·+x n as Hamiltonian. Explicitly, the Hamiltonian vector field X H of LV b (n, k) is given byẋ Setting B (n,k) = 0, one recovers the Hamiltonian system LV(n, k), in particular its Poisson structure Π (n,k) , which was constructed and studied in [6]. Therefore, the system LV b (n, k) is a deformation of the system LV(n, k). We show in the following proposition that the above matrices B (n,k) are the only ones for which the brackets given by (2.2) define a Poisson bracket (on R n ).
Then the brackets given by Proof. Let us denote by Π the Poisson structure defined by A (n,k) , and let us denote the derived Poisson bracket by {· , ·}. We know already from what precedes that if all uppertriangular entries b i,j of B, with j − i / ∈ {m, m + 1} are zero, then Π b is the restriction of a Poisson structure to a Poisson submanifold, hence it is a Poisson structure. We therefore only need to show that if one of these entries b i,j with i < j is non-zero, then Π b is not a Poisson structure. Suppose first that j − i < m and i = 1. Then so that Π b does not satisfy the Jacobi identity. When j − i < m and i = 1 it suffices to replace in the above computation i − 1 by j + 1 to arrive at the same conclusion. Finally, when j − i > m + 1 one replaces in the above computation i − 1 by i + 1 to arrive again at the same conclusion. The rank of Π is the rank of A (n,k) , which is equal to n when n is even and n − 1 when n is odd. Since Π b is obtained by adding constants to the quadratic structure Π, its rank is at least the rank of A (n,k) . However, the rank of Π b is even and bounded by n, so Π b and Π have the same rank, which is 2 n 2 .
The proposition implies that from the above reduction process we get all possible deformations of LV(n, k) obtained by adding to Π (n,k) a constant Poisson structure.
3. The Liouville and superintegrability of LV b (n, 0) We construct in this section enough independent first integrals for the Hamiltonian system LV b (n, 0) to prove its superintegrability and then select from them enough first integrals in involution to prove its Liouville integrability. Notice that since the phase space of LV b (n, 0) is R n and since the Poisson structure on it has rank n or n−1 depending on whether n is even or odd, we need to provide n − 1 independent first integrals to prove superintegrability and n+1 2 independent first integrals (including the Hamiltonian) in involution to prove Liouville integrability. Throughout the section, n is fixed, and k = 0 also, so we will drop from the notations the label (n, 0), except in the statements of the propositions and the theorem.
3.1. First integrals. We first write down the equations for the vector field X H where we recall that H = x 1 + x 2 + · · · + x n and that the Poisson structure Π b = Π (n,0) b is defined by (2.5); the matrix B = B (n,0) has all entries equal to zero, except for b 1,n = −b n,1 , which we will denote in this section by β. Also, the skew-symmetric matrix A = A (n,0) has all its uppertriangular entries equal to (plus!) 1. Therefore, X H is given bẏ (3.1) We construct the first integrals of this system as deformations of the first integrals of LV(n, 0), which were constructed in [18,Prop. 3.1]. We first recall the formulas for these integrals. For 1 ℓ n+1 2 , the following functions F ℓ = F (n,0) ℓ are first integrals of LV(n, 0): More first integrals were constructed by using the anti-Poisson involution ı on R n , defined by ı(a 1 , a 2 , . . . , a n ) := (a n , a n−1 , . . . , a 1 ) , which leaves H is invariant, ı * H := H • ı = H, so that the rational functions G ℓ := ı * F ℓ (ℓ = 1, . . . , n+1 2 ) are also first integrals of LV(n, 0). This yields exactly n − 1 different first integrals, because when n is even, all F ℓ and G ℓ are different, except for F n/2 = H = G n/2 , and when n is odd, all F ℓ and G ℓ are different, except for F (n+1)/2 = H = G (n+1)/2 and F 1 = G 1 . We recall also that the functions F ℓ are pairwise in involution, just like the functions G ℓ , and that all these functions are independent, which accounts for the Liouville and superintegrability of LV(n, 0).
In order to construct from these first integrals of LV(n, 0) first integrals of LV b (n, 0) we use the constant coefficient differential operator D b , which we define for the n × n matrix B corresponding to LV b (n, k) by where we recall that m = n−k−1 and that the indices of B are taken modulo 2m + 1. In view of the conditions on B = B (n,k) (see Proposition 2.2), D b can be written as In the present case, k = 0, and so the skew-symmetric matrix B has only a single non-zero entry above the diagonal, hence Notice that ı * and D b commute, ı * • D b = D b • ı * . As said, we use the operator D b to define some first integrals of LV b (n, 0): we define for 1 ℓ We have used that when the operator D b is applied twice to F ℓ , the result is zero. This follows from the fact that the variables x 1 and x n appear linearly in F ℓ (and hence are absent in D b F ℓ ), as is clear from (3.2). Explicit formulas for the rational functions F b ℓ , with 1 ℓ n−1 2 are given by , if n is odd, Proof. Since ı is an anti-Poisson map which leaves the Hamiltonian H invariant, it suffices to show that the rational functions F b ℓ are first integrals of LV b (n, 0). We do this for odd n, the case of even n being completely analogous. Let 1 ℓ n−1 2 . To prove that F b ℓ is a first integral of (3.1) we show that its logarithmic derivative (log(F b ℓ )) · =Ḟ b ℓ /F b ℓ is zero. Thanks to the particular form of the vector field (3.1), one easily obtains the following two formulas: Summing them up, we find (log(F b ℓ )) · = 0, and hence thatḞ b ℓ = 0. 3.2. Involutivity. We now show that the first integrals F b ℓ of LV b (n, 0) are in involution. For doing this, observe by comparing (3.2) and (3.7) that formally F b ℓ can be obtained from F ℓ by replacing x 1 with x 1 + β/x n . Said differently, if we denote by σ : R n → R n the birational map defined for (a 1 , . . . , a n ) with a n = 0 by σ(a 1 , . . . , a n ) := a 1 + β a n , a 2 , . . . , a n , Since σ * x i = x i for i > 1, this is obvious when 1 < i < j. We therefore only need to verify the formula for i = 1 and j > 1. If 1 < j < n then where we have used in the last step that the functions F ℓ of LV(n, 0) are in involution ( [18,Prop. 3.2]). The fact that the functions G b ℓ are also in involution follows from the fact that ı is an anti-Poisson map of (R n , Π b ).
Notice that although σ is a birational Poisson isomorphism, it is not an isomorphism between the Hamiltonian systems LV(n, 0) and LV b (n, 0) because σ * H = H. In particular, (3.3) does not imply that the rational functions F b ℓ are in involution with the Hamiltonian H, i.e., that they are first integrals of LV b (n, 0); this requires a separate proof, which has been given in Proposition 3.1 above.

3.3.
Integrability. We now prove the Liouville and superintegrability of LV b (n, 0). As we will see, the main result that remains to be proven is that the n − 1 constructed first integrals, to wit the Hamiltonian, the rational functions F b ℓ (with ℓ = 1, . . . , n−1 2 ) and the rational functions G b ℓ (with ℓ = 1, . . . , n−1 2 when n is even and ℓ = 2, . . . , n−1 2 when n is odd) are independent, i.e. have independent differentials on an open dense subset of R n . Since these functions are rational, it suffices to show that their differentials are independent in at least one point of R n .
To see this, we use the fact that the undeformed functions H, F ℓ and G ℓ are independent (at some point P ). Since the deformed functions depend polynomially on the deformation parameter β, they will still be independent at P for β in a small interval, centered at zero. Notice that if we rescale all variables by a factor λ = 0 and rescale β by a factor λ 2 all these functions also get multiplied by a non-zero factor. It follows that the differentials of the deformed functions are independent at P for all values of β. Proof. Recall that a superintegrable system on an n-dimensional manifold is a vector field (Hamiltonian or not), which has n − 1 independent first integrals. As we have constructed precisely this number of independent first integrals for LV b (n, 0), we have proven its superintegrability. For Liouville integrability of a Hamiltonian vector field on an n-dimensional Poisson manifold of rank 2r we need n − r independent first integrals which are in involution. Here, the rank of the Poisson structure Π b is 2 n 2 (see Proposition 2.2) so that we need n − n 2 = n+1 2 such integrals, which is exactly the number of independent first integrals F b ℓ (or G b ℓ ) that we have, and they are in involution by Corollary 3.3. Notice that each of these sets of first integrals contains the Hamiltonian H. Notice also that when n is odd, F b 1 is a Casimir function of Π b .
Example 3.5. For n = 4 and k = 0 the matrices A (4,0) and B (4,0) are given by The corresponding system LV b (4, 0) is given by the formulaṡ and besides the Hamiltonian H = x 1 + x 2 + x 3 + x 4 it has two more independent first rational integrals F and G, namely where ı is the anti-Poisson map defined in (3.3). The above three functions give the superintegrability of the system LV b (4, 0). The rank of the Poisson structure is 4 and each one of the pairs (H, F ) and (H, G) provide the Liouville integrability of LV b (4, 0).

Explicit solutions.
The Hamiltonian vector field X H of LV(n, 0) can be explicitly integrated in terms of elementary functions, as was first shown in [13]. We show that such an integration can also be done for (3.1), the Hamiltonian vector X H of LV b (n, 0). This is most easily done by introducing some new variables, depending linearly on the coordinates x i : for i = 0, 1, . . . , n, let u i := x 1 + x 2 + · · · + x i and notice that u 0 = 0 and u n = H. It is clear that (u 1 , . . . , u n ) defines also linear coordinates on R n , so everything can be easily expressed in terms of the coordinates u i by substituting u i − u i−1 for x i (i = 1, . . . , n). As we will see, this simplifies some of the formulas (for X H , for example) and makes others more complex (the rational integrals, for example). For the proposition which follows, the formulas are the simplest when expressed in the u i variables. First, we need to express LV(n, 0) in terms of the coordinates u 1 , . . . , u n . The simplest way to do this is to first compute Π b in terms of the u i variables.
Since H = u n , we can compute X H as {· , u n } b , which takes in view of the above formulas the following simple, decoupled form: which can easily be integrated, for any initial condition. We describe the integration in a geometrical language, which will be useful when we use it in Section 5. For any point P ∈ R n , we can consider the integral curve of X H , starting from P , which we will denote by γ P . Usually, the domain of an integral curve is taken to be an interval, but in the present case we will take it to be all of R minus a discrete subset. On the one hand, it is natural to do this because in the case of X H the solutions are precisely defined on such a set. On the other hand, the systems LV b (n, k) can equally be defined on a complex phase space C n and then the integral curves, with complex time, are defined for all of C, minus a discrete subset; the domain of the real integral curves which we consider is just the real part of this complex subset. Since it is convenient to express the integral curves in terms of coordinates (here the u i coordinates) we will write, once P has been fixed, u i (t) for u i (γ P (t)). The u-coordinates of P will be denoted (P 1 , . . . , P n ), so P i = u i (P ) for i = 1, . . . , n − 1 and P n = u n (P ) = H(P ).
Proposition 3.6. Let P be any point of R n and let γ P denote the integral curve of (3.10), which is LV b (n, 0), expressed in the u i variables, starting from P . Denote by h the value of the Hamiltonian at P , i.e., h = H(P ) = u n (P ). Let ∆ 0 be a square root of h 2 + 4β, which may be real or imaginary. Then, for i = 1, 2, . . . , n − 1,

The Liouville and noncommutative integrability of LV b (n, k)
In this section, we generalize the results of Section 3 on the integrability of LV b (n, 0) to the case of LV b (n, k), where k ∈ N satisfies 2k + 1 < n, but is otherwise arbitrary. We do not treat here the case of n = 2k + 1 because we have already established the Liouville integrability of LV b (2k + 1, k) in [7]. We show in this section that if 1 < 2k + 1 < n then LV b (n, k) is on the one hand Liouville integrable, and on the other hand is non-commutatively integrable of rank k +1. We start by recalling the definition of non-commutative integrability (see [16,14]), which we specialize to R n . (2) For P in a dense open subset of R n : Then the triplet (R n , Π, F) is called a non-commutative integrable system of rank r.
The classical case of a Liouville integrable system corresponds to the particular case where r is half the (maximal) rank of Π; this implies that all the functions f 1 , . . . , f s are pairwise in involution. The case of a superintegrable system corresponds to r = 1; in this case, condition (1) just means that one has n − 1 first integrals, while the second condition in (2) is trivially satisfied: superintegrability means, as recalled in the previous section, that one has n − 1 independent first integrals.
In order to establish Liouville and non-commutative integrability in Section 4.3 below, we first construct a set of polynomial first integrals for LV b (n, k), which are pairwise in involution, and then we construct a set of rational first integrals for LV b (n, k), which are also pairwise in involution. This will be done in the two subsections which follow. Throughout the section, we suppose that 1 < 2k + 1 < n and that B = B (n,k) is a skew-symmetric n × n matrix such that (2.5) defines a Poisson structure on R n . Recall that this means that the uppertriangular entries b i,j of B with j − i / ∈ {m, m + 1} are zero.
4.1. The polynomial integrals. Recall from Section 2.2 that the systems LV b (n, k) are obtained by reduction from the systems LV b (2m+1, m), where m := n−k−1 > k, where the last inequality comes from our assumption n > 2k + 1. Recall also that in order to do this reduction, one supposes that the last 2m+1−n rows and columns of the (2m+1)×(2m+1) matrix B are zero, so that B can be viewed as an n × n matrix by removing these zero rows and columns (see Proposition 2.1). Since LV b (2m + 1, m) is Liouville integrable, with m + 1 independent polynomial integrals, whose formulas are recalled below, we obtain by reduction a set of first integrals of LV b (n, k), which are automatically in involution with respect to the reduced Poisson structure, which is by definition the Poisson structure Π One has however to be careful with the independence of the reduced integrals, for example some of these reduced integrals are zero! Moreover, since n > 2k+1, more integrals are needed for integrability, as we will see.
Let us first recall the formulas for the (polynomial) first integrals of LV b (2m + 1, m). One method of constructing them is as coefficients of the characteristic polynomial of the Lax operator L(λ) := X + λ −1 ∆ + λM , which we recalled in (2.4). It is a classical fact that the coefficients of the characteristic polynomial of a Lax operator yields first integrals for any Lax equation in which it appears [14,Sect. 12.2.5]. For L(λ), the following expansion of its characteristic polynomial was obtained in [7,Prop. 8]: (4.1) Thus, the polynomials K b i in this expansion are first integrals of LV b (2m + 1, m). Setting the deformation parameters equal to zero, one recovers the first integrals of LV(2m + 1, m), which were first constructed by Bogoyavlenskij [4] and Itoh [12]. We construct k + 1 first integrals of LV b (n, k) by setting, for i = 0, . . . , k, where the notation introduced by the latter equality is a convenient shorthand. By construction, these polynomials are first integrals of LV b (n, k) and they are in involution. Also, K 1 = H and K i is of degree 2i + 1 for i = 0, 1, . . . , k.
We give an alternative description of the first integrals (4.2) as deformations of the polynomial first integrals of LV(n, k). On the one hand, this description will be important for showing the independence of these first integrals, and on the other hand it will provide information about the structure of these integrals, which we will use to prove some of their properties (involutivity, for example).
It was shown in [7, Prop. 9] that the integrals K b i of LV b (2m + 1, m) can be obtained using the D b operator (see (3.4)) by the following formula, valid for i = 0, 1, . . . , m: Let us show that D b commutes with restriction to R n . Let F be a smooth or rational function on R 2m+1 . In view of the conditions on B, the operator D b is given by (3.5) and we see that D b does not involve derivation with respect to any of the variables x n+1 , x n+2 , . . . , x 2m+1 (recall that n = m + k + 1), so D b commutes with restriction to the subspace R n of R 2m+1 , which is defined by x n+1 = x n+2 = · · · = x 2m+1 = 0. It follows that This shows that the polynomial first integrals K For later use, we quickly recall from [6] a combinatorial formula for K (n,k) i . Let m = (m 1 , m 2 , . . . , m 2i+1 ) be a 2i + 1-tuple of integers, satisfying 1 m 1 < m 2 < · · · < m 2i+1 n. We view these integers as indices of the rows and columns of A (n,k) : we denote by A (n,k) m the square submatrix of A (n,k) of size 2i + 1, corresponding to rows and columns m 1 , m 2 , . . . , m 2i+1 of A (n,k) , so that (A (n,k) m ) s,t = (A (n,k) ) ms,mt , for s, t = 1, . . . , 2i + 1 . Letting One immediate consequence is that every variable x j has degree at most one in K (n,k) i .

4.2.
The rational integrals. We now construct a set of rational integrals of LV b (n, k). In order to follow some of the more technical arguments in this subsection, the reader is advised to already take a look at Section 4.4 below, where explicit formulas for a few examples are given.
Recall that we assume in this section that k > 0 and that n > 2k + 1. We define the rational integrals of LV b (n, k) as deformations of the rational integrals of LV(n, k), which were first constructed in [6]. We first recall the definition of the latter integrals as pullbacks of the rational first integrals F (n,0) ℓ and G (n,0) ℓ of LV(n − 2k, 0), which we recalled in Section 3.1. Consider the polynomial map φ k : R n → R n−2k , defined by φ k (a 1 , a 2 , . . . , a n ) := a 1 a 2 . . . a k (a k+1 , a k+2 , . . . , a n−k )a n−k+1 . . . a n−1 a n .
If we denote the standard coordinates on R n by x 1 , . . . , x n , and on R n−2k by y 1 , . . . , y n−2k , then φ * y i = x 1 x 2 . . . x k x k+i x n−k+1 . . . x n for i = 1, . . . , n − 2k. It was shown in [6] that for ℓ = 1, 2, . . . , n+1 2 − k, the rational func- are first integrals of LV(n, k) and that the integrals F (n,k) ℓ are pairwise in involution with respect to {· , ·} (n,k) , just like the integrals G (n,k) ℓ . Setting s := 2ℓ − 1 when n is odd and s := 2ℓ when n is even, so that s and n have the same parity, one computes easily from (3.2) that When ℓ = n+1 2 − k, the last product in this expression reduces to 1 and so F (n,k) ℓ is actually a polynomial function. Since some of the arguments below which depend on the structure of the integrals fail for the polynomial first integrals, we will exclude these integrals in this section, and so we will throughout this section only consider the rational integrals F (n,k) ℓ , with ℓ = 1, 2, . . . , n−1 2 − k. Notice also that every variable or its inverse appears precisely once in (4.5), with the k variables x 1 , . . . , x k and the k +1 variables x n−k , . . . , x n appearing linearly. This property will be important in what follows.
We construct the rational first integrals of LV b (n, k) as deformations of the integrals F (n,k) ℓ by using the operators D b (see (3.4) or (3.5)): for ℓ = 1, 2, . . . , n−1 and similarly for G . The present subsection is devoted to the proof of the following theorem, which says that the deformed integrals F  For the proof of Theorem 4.2, we need some extra notation. Since throughout this subsection n and k are fixed, we will until the rest of the subsection drop (n, k) from the notation, writing F b ℓ for F (n,k),b ℓ , writing B for B (n,k) , and so on. We will need a specific ordering of the entries of the matrix B. Therefore, we label the parameters of B with single indices as follows (recall that B is skew-symmetric and that its non-zero uppertriangular entries are at positions (i, j) with j − i = n − k or j − i = n − k − 1): Expressed in terms of a formula, the parameters b p+1 , b p+2 , . . . , b 2k+1 equal to zero, is denoted by Π (p) . In particular, Π b = Π (2k+1) . The associated Poisson bracket is denoted by {· , ·} (p) . We also associate to p the following constant coefficient differential operator .

(4.6)
It is clear that D b = D (2k+1) + D (2k) + · · · + D (2) + D (1) and, since these operators commute, that We could of course consider any alternative order of the operators, but we will use the above one for some reason which will become clear later. Finally, for any smooth or rational function F on R n we set F (p) := e D (p) F (p−1) for 1 p 2k + 1, and F (0) := F. With this notation, A crucial fact which we will use is that the partially deformed integral F (p) ℓ is the pullback of F (p−1) ℓ by the birational Poisson map σ p , defined for 1 i n by if p is odd, if p is even. (4.7) Notice that independently of whether p is even or odd.
when p is odd and is independent of when p is even. We do this for p odd, i.e., we show that when p is odd, where E 1 is independent of x n− p−1 2 and of x k− p−1

2
, while E 2 is independent of x n− p−1

2
. According to the explicit formula (4.5), we can write where we have used in the last step that D 2 (p) F ℓ = 0, which is a consequence of the fact that F ℓ depends linearly on x k+1− p−1 2 and on x n− p−1 2 , which are the variables with respect to which D (p) differentiates (see (4.6)). Since, and of x k− p−1

2
, and E 2 is independent of x n− p−1 2 . This shows our claim when p is odd. The proof in case p is even is similar. We use the obtained formula (4.9) to show that F for any p. According to (4.9), we can write when p is even and is independent of when p is odd. Therefore, on the one hand, while on the other hand, We have used in (⋆) that σ * p (E 1 ) = E 1 and that σ * p (E 2 ) = E 2 , which hold because E 1 and E 2 are independent of the only variable which is not fixed by σ * p .
We next show that the maps σ p are Poisson maps with respect to the appropriate Poisson structures on R n . Proof. We give the proof in case p is even. Then (4.7) simplifies to We need to show that σ * as was to be shown. Suppose now that 1 i = n + 1 − p 2 < j. Notice that, in this case, {x i , x j } (p) = {x i , x j } (p−1) = +x i x j , with a plus sign. Then where we have used that {x i+k−n , x j } (p) = −x i+k−n x j , with a minus sign because i < j. Finally, suppose that 1 i < j = n + 1 − p 2 and notice that where the + sign corresponds to the case i > k + 1 − p 2 and the − sign to the case i k + 1 − p 2 . Clearly, this gives the same result as in (4.10). This shows that σ p is a Poisson map.
and so, since the undeformed rational integrals F ℓ = F We will next show that that the deformed integrals F b ℓ are first integrals of LV b (n, k). To do this, we first prove the following lemma. Proof. We prove by recursion on p that x i , F where we used the recursion hypothesis in the last step. Suppose now that σ * p (x i ) = x i . If p is even, this means that i = n + 1 − p 2 , and so where we used again the recursion hypothesis twice: we could do so because i + k − n k so that i + k − n / ∈ {k + 1, k + 2, . . . , k + s} . If σ * p (x i ) = x i and p is odd, then i = 1 + k − p−1 2 , so that as before.
Using the lemma and the fact that F ℓ is a first integral of LV(n, k) (see [6]), we show that F b ℓ is a first integral of LV b (n, k). As before, we show by recursion that F = 0 for some p 1. Then k when p is even and t = n − p−1 2 n − k when p is odd; in either case, t / ∈ {k + 1, k + 2, . . . , k + s}, which proves the last equality. We conclude that F b ℓ , H b = 0, which is the first statement of Theorem 4.2.

Integrability.
We have now most ingredients to state and prove the Liouville and non-commutative integrability of LV b (n, k), where we recall that k > 0 and 2k + 1 < n. Since in this subsection (n, k) is fixed we will again drop (n, k) from the notation, except in the statement of the propositions and of the theorem. We have constructed in Section 4.1 a set of polynomial integrals for LV b (n, k) and in Section 4.2 a set of rational integrals F ℓ . We first show that these polynomial integrals are in involution with these rational integrals. To do this, we need a property of the polynomial integrals, which we first define.
Definition 4.6. A polynomial function K on R n is said to be (n, k)-admissible if (1) K is of degree at most one in each of its variables x j ; (2) K can be written (uniquely) as K = LK ′ + K ′′ , where K ′′ is independent of x k+1 , . . . , x n−k and L is the sum of these variables, L = x k+1 + x k+2 + · · · + x n−k .
A key property of the polynomial Hamiltonians is that they are (n, k)admissible: Proposition 4.7. For i = 0, 1, . . . , k, the polynomial first integral K (n,k),b i is (n, k)-admissible.
Proof. As said, we write in the proof K i for K when k < m ′ i+1 < n − k + 1. According to the formula (4.4) for K i this means that when some term of K i contains a variable x j with k + 1 j n − k, then it contains also a similar term with x j replaced by any x l with k+1 l n−k. Considering the sum of these substitutions yields a polynomial which is divisible by L. Therefore, K i is (n, k)-admissible. Let us show that if for some p 1, K showing that K (2) i is also (n, k)-admissible. The proof for p = 2k + 1 is very similar, since D (2k+1) differentiates with respect to the variables x 1 and x n−k .
We are now ready to show that every polynomial integral is in involution with every rational integral. Proof. In view of Proposition 4.7, we can write K b i = LK ′ + K ′′ where K ′ and K ′′ are independent of x k+1 , . . . , x n−k . Using Lemma 4.5 twice, The Hamiltonian H is of course also (n, k)-admissible, H = L+H ′′ , with H ′′ independent of the variables x k+1 , . . . , x n−k . Using that F ℓ is a first integral (Theorem 4.2) and Lemma 4.5, we can conclude that Combining the results obtained in this section, we can state and prove the main theorem on the integrability of the systems LV b (n, k), with n 2k + 1. We denote, in that order, by H n−k−2 the following first integrals: Proof. We first consider (1). We have already checked the first item of Definition 4.1, namely that the k + 1 polynomials K i are first integrals of LV(n, k), and are in involution with both the polynomial and rational integrals (Section 4.1 and Proposition 4.8). We need to check the second item which says that the differentials of these first integrals are independent on a dense open subset of R n , and similarly for the Hamiltonian vector fields associated to the polynomial integrals. To do this we use the fact that the undeformed first integrals have this property, as they define a noncommutative integrable system of rank k+1 (see [6,Theorem 1.1]). Since all integrals are rational functions and since the Poisson structure is polynomial, it suffices to prove that the differentials (resp. vector fields) are independent at some point. The argument is the same as the one used in Section 3.3 to derive the independence of the integrals of LV b (n, 0) from the independence of the integrals of LV(n, 0): since the property is true at some point P when all parameters are zero, it is still true on a neighborhood of P for small values of the parameters; by rescaling the variables and parameters, one finds that at P the property is true for all values of the parameters. This proves (1). We now consider (2), the Liouville integrability. Since the rank of Π b is n when n is even and n − 1 when n is odd, we need n/2 independent integrals in involution when n is even and n+1 2 when n is odd. Clearly, the above list in (2) contains k + s = n+1 2 functions, which is the right number, we know that they are pairwise in involution, and by the above argument they are independent. So they define a Liouville integrable system.
Item (1) in the theorem takes a slightly different form when n = 2k + 1. The constructed first integrals are then polynomial and they define a noncommutative integrable system of rank k, which is equivalent to saying that it is Liouville integrable, which is stated in (2), and was already proven in [7]. The reason of this drop in the rank of the non-commutative integrability when n = 2k + 1 is because, even though we have k + 1 polynomial integrals that are in involution with all integrals, like the general case of the LV b (n, k) systems, now one of these k + 1 polynomial integrals is a Casimir and in order to establish the condition (2) of Definition 4.1 one has to exclude the Casimir from the first set of integrals.   1) and B (4,1) are given by The corresponding system LV b (4, 1) is given bẏ Besides the Hamiltonian H = x 1 +x 2 +x 3 +x 4 it has an additional polynomial integral x 1 which is easily seen to be a (4, 1)-admissible polynomial. The above two polynomials give the Liouville integrability of the system LV b (4, 1) which coincides in this case with the non-commutative integrability of rank k +1 = 2 just like in all LV b (2k +2, k) systems.
Example 4.11. We now consider the case n = 7 with k = 1. The matrix A := A (7,1) is the skew-symmetric Toeplitz matrix with first line (0, 1, 1, 1, 1, 1, −1) and B := B (7,1) is the skew-symmetric matrix whose only non-zero upper triangular entries are b 1,6 = b 3 , b 1,7 = −b 2 and b 2,7 = b 1 . The corresponding system LV b (7, 1) is given by the equationṡ Besides the Hamiltonian H = x 1 + x 2 + · · · + x 7 , the system LV b (7, 1) has one more independent polynomial first integral K 1 , given by which is a (7, 1)-admissible polynomial. It has also three rational first integrals given by and G 2 = ıF 2 . The rank of the Poisson structure Π (7,1) b is 6 and F 1 is a Casimir, invariant under ı * . It can be seen that the above integrals are obtained from the undeformed ones (obtained by setting the parameters equal to zero), by applying on them the operator e D b which now becomes The system LV b (7, 1) is non-commutative integrable of rank 2 with first integrals H, K 1 , F 1 , F 2 , G 2 and is also Liouville integrable with first integrals H, K 1 , F 1 , F 2 or H, K 1 , F 1 , G 2 .

Discretization of LV b (n, 0)
In this section we construct a family of discretizations of LV b (n, 0). They are obtained from a discrete zero curvature condition, which is the compatibility condition of a linear system LΨ = λΨ,Ψ = N Ψ, where L is the Lax matrix of LV(n, 0), which appears in (2.4). We prove that an important class of these discretizations, which includes the Kahan (also called Kahan-Hirota-Kimura) discretization of LV(n, 0) has the following integrability properties: it has the rational integrals of LV(n, 0) as invariants, and so it is both Liouville and superintegrable; also it has an invariant measure.
Throughout this section, (n, k) = (n, 0) is fixed and so we will drop (n, 0) from the notation for the invariants, the Poisson structure, and so on. Also, since we have in this case only one parameter b 1,n , we will denote it by β, as we did in Section 3.

Preliminaries.
We first recall a few basic definitions and properties of discrete maps and their integrability. By a discrete map of R n we mean an algebra homomorphism Φ : R(x 1 , x 2 , . . . , x n ) → R(x 1 , x 2 , . . . , x n ), where x 1 , . . . , x n are as elsewhere in this paper the natural coordinates on R n . Such a map is the pullback of a unique rational map φ : R n → R n , i.e., for any rational function F , one has Φ(F ) = φ * (F ) = F • φ. We will also use the convenient abbreviationsF for Φ(F ). Similarly for a matrix P = (p i,j ) whose entries are rational functions of R n , we will writeP for the matrix (p i,j ).
When R n is equipped with a Poisson structure Π, then saying that Φ is a homomorphism of Poisson algebras is tantamount to saying that φ is a Poisson map; we will simply say that Φ preserves the Poisson structure Π. Also, on R n we have a natural n-form, dx 1 ∧ · · · ∧ dx n which allows us to identify rational measures with rational n-forms and with rational functions. We will say that Φ is measure preserving, with preserved measure F , if it preserves the n−form F dx 1 ∧ dx 2 ∧ . . . ∧ dx n in the sense that A rational function F is called an invariant of Φ ifF = F . We also recall the definition of an integrable map [19].
(1) The map Φ is Liouville integrable if there exist n − r functionally independent invariants of Φ, which are in involution with respect to a Poisson structure Π, where r is half the rank of Π. (2) The map Φ is superintegrable if it has n − 1 functionally independent invariants and is measure preserving.
In order to simplify some of the computations below, we introduce a few more notions and notations which are related to the symmetries of the Lax matrix L of LV(2m + 1, m). Recall that is the square matrix of size 2m + 1, where The entries of the above matrices are functions of x 1 , x 2 , . . . , x 2m+1 where all the indices are considered modulo 2m + 1. It is a τ -circulant matrix, in the sense of the following definition: Definition 5.2. Let p, q ∈ N and let τ be a map τ : R p → R p of order q. Let C = (c i,j ) be an q × q matrix whose elements are functions on R p . Then we call C a τ -circulant matrix if it has the following property: c i+1,j+1 = τ * c i,j , for all i, j = 1, 2, . . . , q .
If v is the first line of C we will also denote C by C(v).
Taking for τ the identity map, one recovers the definition of a circulant matrix. As in the case of a circulant matrix, it is clear that a τ -circulant matrix is determined by its first line. It is easy to see that a matrix C is τ -circulant if and only if M CM −1 = τ * (C), where M is the circular permutation matrix, defined in (5.2), and τ * (C) is the matrix obtained by applying τ * to all entries of C. As an immediate consequence, we see that the product of two τ -circulant matrices is τ -circulant; to compute the product of two such matrices one has to compute only the first line of the product, a property which we will find useful.
In order to see that the Lax operator L is τ -circulant, let us consider R 4m+2 with coordinates x i , b m+i,i for i = 1, 2, . . . , 2m + 1. The map τ from R 4m+2 to itself is defined by τ * (x i ) = x i+1 and τ * (b m+i,i ) = b m+i+1,i+1 for i = 1, 2, . . . , 2m + 1. Since all indices are considered modulo 2m + 1, the map τ is of order 2m + 1. To see that L is τ -circulant, we show that the matrices which define L are τ -circulant, which is clear from the following formulas:

5.2.
Discrete maps from a linear problem. Recall that LV b (n, 0) is by definition a reduction of LV b (2m+1, m), with m := n−1. The discrete maps of LV(n, 0) which we will consider below are in the same sense reductions of discrete maps of LV b (2m + 1, m), which we first construct. We construct a discrete map Φ by considering the compatibility conditions of the linear system LΨ = λΨ,Ψ = N Ψ , where L is the Lax matrix of LV b (2m + 1, m), recalled above, and Ψ is an n-dimensional eigenvector of L. Recall thatΨ is the vector Ψ with Φ applied to its entries. The (2m + 1) × (2m + 1) matrix N is defined as where K i,j := δ i,j+m and D i,j := δ i,j d i for some functions d i that will be determined from the compatibility condition of (5.3), which readsLN = N L.
Since N is invertible, it means thatL = N LN −1 and therefore the coefficients of the characteristic polynomial of L, which are rational functions on R 2m+1 , are invariants of Φ. The above ansatz for N was taken so that N LN −1 equals L at the entries with constant values. Therefore, the compatibility conditionLN = N L reduces to a system of equations for thex i and d i variables, which we make explicit in the following proposition: Proposition 5.3. The compatibility conditionLN = N L of the linear system (5.3) is equivalent to the following system of equations: 5) for i = 1, 2, . . . , 2m + 1 = 2n − 1.
Proof. Notice first that N is τ -circulant, if we extend τ * to the variables d i , by setting τ * d i := d i+1 . The compatibility conditionLN = N L amounts therefore to the equality of the first rows ofLN and N L, which are respectively given by where the non-zero components of the above vectors appear at the positions 1, 2, m + 2 and m + 3. From the equality of these first lines, we get which is (5.5) for i = 1. The other equations follow from it by τ -circularity.
We now reduce these equations to LV b (n, 0), setting x i =x i = 0 for i = n + 1, n + 2, . . . , 2m + 1 and b m+i,i = 0 for i = 2, 3, . . . , 2m + 1, where we recall that m = n − 1 and that we denote the single parameter b 1,n of LV(n, 0) as b 1,n = β. The system (5.5) is then transformed to the following one: where the first three equation are instances of the first equation in (5.5) and the last three equations of the second one. Before solving the above system, we recall from Section 3.4 the alternative coordinates u 1 , . . . , u n for R n , in which the system LV b (n, 0) completely separates. They are defined by u i = n j=1 x j for all i = 0, 1, 2, . . . , n. For i = n, u n is just the Hamiltonian, Proposition 5.4. For any rational function R ∈ R(x 1 , . . . , x n ), different from the n functions u i − H, with i = 1, . . . , n, the reduced compatibility equations (5.6) have a unique solution forx 1 , . . . ,x n and for d 2 , . . . , d 2n−1 , with d 1 = R. It is given bỹ  Proof. We first show how the third and fourth equations in (5.7) are derived from (5.6). The last equation is obtained from the third equation in (5.6): for i = 1, . . . , n − 1, where we have used that, by periodicity, d 2n = d 2m+2 = d 1 = R. In order to derive the third equation in (5.7), one first uses the first three equations in (5.6) to substituex i (i = 1, . . . , n) and x i (i = 2, 3, . . . , n) in the fourth and fifth equations in (5.6), to obtain, in that order, The first equation in (5.8) says that d i+1 d n+i is independent of i for i = 1, . . . , n − 2, while the second equation says that this constant value is equal for i = 1, . . . , n − 2. By our assumption on R, the d n+i = R + H − u i with i = 1, . . . , n − 1 are all different from zero, so that we can divide (5.9) by d n+i . It yields the third equation in (5.7). This shows that (5.7) is the only possible solution for (5.6) with d 1 = R. That it is indeed a solution is easily verified by substituting the formulas (5.7) in (5.6).
We now define a discrete map using the solution of (5.6) given in the previous proposition. Let R be a rational map, with R = u i − H for all i = 1, 2, . . . , n − 1, and let Φ R be the discrete map x i →x i , defined by the formulas , i = 1, 2, . . . , n − 1 , where we have set u 0 = 0. Using the first n equations in (5.6), we get u i = d i+1 − R and therefore the map is given in terms of the coordinates u i byũ n = u n andũ Since the map Φ R is by construction isospectral, it has the coefficients of the characteristic equation of the Lax matrix L as invariants. However, as we noted just after equation (4.4), we get in this way only one invariant, namely the Hamiltonian H. We show in the following proposition that Φ R also preserves the rational first integrals of LV b (n, 0). Proposition 5.5. Let P be any point of R n for which Q := φ R (P ) is defined. Then Q belong to the integral curve of the continuous system LV(n, 0) starting at P . In particular, the discrete map Φ R preserves all the integrals of LV b (n, 0).
Proof. Since Φ R is a rational map and the integrals of LV b (n, 0) are rational functions, it suffices to show that for a generic P of R n for which Q := φ R (P ) is defined, Q belongs to the integral curve of the continuous system LV(n, 0) starting at P . We use the notation of the proof of Proposition 3.6: we denote by γ P the integral curve of (3.10) starting from P , and we write u i (t) = u i (γ P (t)). We denote by h the value of the Hamiltonian at P and by ∆ 0 a square root of h 2 + 4β, which may be real or imaginary. Also, let r 0 denote the value of R evaluated at P and Q = (Q 1 , Q 2 , . . . , Q n ).
It is clear from the above that we only need to show that for each P such that Q is defined there exists a t, depending only on P , such that Q i = u i (t) for all i = 1, 2, . . . , n − 1. It is also clear that we may consider our system LV b (n, 0) living on C n and therefore the integral curves are defined on all of C minus a discrete set (see Section 3.4 for details and comments).
We only need to consider the case that ∆ 0 = 0. In this case the solution of LV b (n, 0), for gives Q i = u i (t) for all i, as it can be seen by comparing the formulas (5.11) and the explicit solution of LV b (n, 0) given in Proposition 3.6.

5.3.
Integrable discretization of LV b (n, 0). For a general rational function R, the map Φ R = φ * R cannot be expected to have any integrability properties. We establish in this subsection a few results under the assumption that R is a first integral of LV(n, 0), or under the stronger hypothesis that R depends on H only. We first prove that, under these conditions, Φ R is birational.
Proposition 5.6. Suppose that R is a first integrals of LV b (n, 0). Then φ R is a birational map, so that Φ R is an algebra isomorphism.
Proof. Let R be as announced, so thatR = R, in view of Proposition 5.5. Let Ψ be the involutive algebra homomorphism, defined by Ψ(x i ) :=x n+1−i and Ψ(d i ) := d 2n+1−i for i = 1, . . . , n; notice that d 1 is fixed under Ψ. If we apply Ψ to the reduced compatibility equations (5.6) we get the same set of equations: the first and third equations are permuted, as well as the last ones, while the other two are unchanged. Since we know from Proposition (5.4) that given d 1 := R the reduced compatibility equations (5.6) have a unique solution forx 1 , . . . ,x n and for d 2 , . . . , d 2n−1 , in terms of x 1 , . . . , x n , this means given d 1 :=R = R, they also have a unique solution for x 1 , . . . , x n and for d 2 , . . . , d 2n−1 , in terms ofx 1 , . . . ,x n . Therefore, the map Φ R , defined by the solutions of the system (5.6), is birational.
We now prove that Φ R is, under the same assumption on R, measure preserving.
Proposition 5.7. Suppose that R is a first integral of LV b (n, 0). Then the discrete map Φ R preserves the rational n-form Proof. We need to show that Since the coordinate change between the coordinates u i and x i have triangular form, and since the functionsũ i depend in the same way on thex i , i.e.,ũ i = i j=1x i , we have that where the above determinants are the Jacobian determinants of these two transformations. This implies that we need to show that du 1 ∧ du 2 ∧ · · · ∧ du n x 1 x 2 . . . x n + βx 2 x 3 . . . x n−1 = dũ 1 ∧ dũ 2 ∧ · · · ∧ dũ ñ x 1x2 . . .x n + βx 2x3 . . .x n−1 .
We assume for the moment that R is any function and we denote by R i the partial derivatives ∂R ∂u i and by u i,j the partial derivatives ∂ũ i ∂u j . The explicit formulas forũ i give, for any i = 1, 2, . . . , n − 1 and any j = i, n, that u i,j = R jui (R + H + u i ) 2 .
(5.12) Also, u i,i = R iui +RH+R 2 −β (R+H+u i ) 2 for all i = 1, 2, . . . , n − 1. For the partial derivatives of theũ i with respect to u n , as we will see, the explicit formulas are not important, we will only use the fact thatũ n = u n and therefore u n,j = δ n,j , which holds for any function R. Differentiating (5.11) and rearranging, we get dũ i = n j=1 u i,j du j , i = 1, 2, . . . , n .
Taking the wedge product of the above n equations we obtain dũ 1 ∧ dũ 2 ∧ · · · ∧ dũ n = det(U ) du 1 ∧ du 2 ∧ · · · ∧ du n , where U = (u i,j ). The last line of U is the vector (0, 0, . . . , 1) and therefore expanding the determinant det(U ) with respect to the last line we get det(U ) = det(V ) where V is the minor of U obtained by removing its last row and last column. According to the formulas (5.12) the determinant of the (n − 1) × (n − 1) matrix V has the following form: where s = n−1 j=1 (R + H + u i ) 2 and r = RH + R 2 − β. The above matrix is written as W + rI n−1 , where I n−1 is the (n − 1) × (n − 1) identity matrix and W has n equal lines. This means that W has only two eigenvalues λ 1 and λ 2 . The first one is λ 1 = n−1 j=1 R juj , which is of multiplicity 1 and the other one is λ 2 = 0 of multiplicity n − 2. In the particular case where R is a first integral of LV b (n, 0), the eigenvalue λ 1 reduces to zero (sinceu n = 0). This shows that, in that case, Therefore, what we need to show is that 1x2 . . .x n + βx 2x3 . . .x n−1 x 1 x 2 . . . x n + βx 2 x 3 . . . x n−1 .
A comparison with the explicit formulas (5.10) gives that .
To complete the proof, it remains to be shown that .
This can be done by substituting the formulas forx 1 andx n , given in (5.10).
In order to preserve the Poisson structure, one needs stronger conditions on R, as given in the following proposition: Proposition 5.8. Suppose that R is a rational function of the Hamiltonian H. Then the map Φ R preserves the Poisson structure Π b .
Proof. We give the proof using the coordinates u i (see Section 3.4, in particular the formulas (3.9) for the Poisson structure in terms of these coordinates). According to (3.9) we need to show that {ũ i ,ũ j } b =ũ i (ũ j −ũ i ) and that {ũ ℓ , u n } b =ũ ℓ (ũ n −ũ ℓ ) + β for 1 i < j < n and 1 ℓ < n.
The derivatives ofũ i , for any i < n, with respect to u i and u n = H are where R H = dR dH . We then have, for all i, j and ℓ as above, which establishes the required equalities, sinceũ n = u n = H.
The above propositions lead to the following theorem.
Theorem 5.9. Let R be a rational function, depending on the Hamiltonian H only. Then the discrete map Φ R of LV(n, 0) has the following properties: (1) It is birational; (2) It preserves the Poisson structure Π b ; (3) It is measure preserving: it preserves the volume form Ω b ; (4) It is Liouville integrable with H and the rational functions F b ℓ as invariants; (5) It is superintegrable with H and the rational functions F b ℓ and G b ℓ as invariants.
Under the weaker hypothesis that R depends only on the invariants of LV(n, 0), items (1), (3) and (5) still hold, but (2) and (4) may not hold. 5.4. Kahan discretization of LV b (n, 0). In this subsection we consider the Kahan discretization of the systems LV b (n, 0). We show that the Kahan map is of the form Φ R , for a specific choice of the rational function R, depending on the Hamiltonian H only, and so all integrability properties that we have seen in Theorem 5.9 hold for the Kahan map as well.
We first define the Kahan map for LV b (n, 0). Since the Kahan discretization commutes with any linear change of variables, we can do the Kahan discretization in the u i coordinates, instead of the x i coordinates, i.e., apply it on the vector field (3.10). Following the recipe [5], we obtain for the Kahan discretization with step size 2ǫ the following system of equations: u i − u i = ǫu i (H −ū i ) + ǫū i (H − u i ) + 2β, i = 1, 2, . . . , n − 1, (5.14) where we have used that H = u n . Since H is a linear first integral of LV b (n, 0), it is an invariant for the Kahan map. The system (5.14) is diagonal with solutionū i = (1 + ǫH)u i + 2ǫβ 1 − ǫH + 2ǫu i , i = 1, 2, . . . , n − 1 (5.15) andū n = u n . This defines the Kahan map. Comparing the formulas (5.15) and (5.11) it is clear that the Kahan map is of the form Φ R , with Notice that R depends on H only. Therefore, we get by Theorem 5.9 the following results on the Kahan discretization of LV b (n, 0), which generalize the results on the integrability results on the Kahan discretization of LV(n, 0), which were first established in [18]: Theorem 5.10. The Kahan map of LV(n, 0) has the following properties: (1) It is birational; (2) It preserves the Poisson structure Π b ; (3) It is measure preserving: it preserves the volume form Ω b ; (4) It is Liouville integrable with H and the rational functions F b ℓ as invariants; (5) It is superintegrable with H and the rational functions F b ℓ and G b ℓ as invariants.
As a byproduct of our analysis, we find that the Kahan map of LV b (n, 0) arises as the compatibility conditions of a linear system. It would be interesting to see if there are other examples where the Kahan map is of this form, as it links the Kahan map to isospectrality, so it may have non-trivial applications to the study of the integrability of the Kahan map of other integrable systems.