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JGM

Let $X$ be a Hamiltonian vector field defined on a symplectic manifold $(M,\omega)$, $g$ a nowhere vanishing
smooth function defined on an open dense subset $M^0$ of $M$. We will say that the vector field $Y=gX$ is
\emph{conformally Hamiltonian}. We prove that when $X$ is complete, when $Y$ is Hamiltonian with respect to
another symplectic form $\omega_2$ defined on $M^0$, and when another technical condition is satisfied,
then there is a symplectic diffeomorphism from $(M^0,\omega_2)$ onto an open subset of $(M,\omega)$,
which maps each orbit to itself and is equivariant with respect to the flows of the vector fields
$Y$ on $M^0$ and $X$ on $M$. This result explains why the diffeomorphism of the phase space of the Kepler problem restricted
to the negative (resp. positive) values of the energy function, onto an open subset of the cotangent bundle
to a three-dimensional sphere (resp. two-sheeted hyperboloid), discovered by Györgyi (1968) [10],
re-discovered by Ligon and Schaaf (1976) [16], is a symplectic diffeomorphism.
Cushman and Duistermaat (1997) [5] have shown that the Györgyi-Ligon-Schaaf diffeomorphism is characterized
by three very natural properties; here that diffeomorphism is obtained by composition of the diffeomorphism
given by our result about conformally Hamiltonian vector fields with a (non-symplectic)
diffeomorphism built by a variant of Moser's method [20]. Infinitesimal
symmetries of the Kepler problem are discussed, and it is shown that their space is a Lie algebroid
with zero anchor map rather than a Lie algebra.

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