doi: 10.3934/dcdss.2020070

On the geometry of twisted prolongations, and dynamical systems

Dipartimento di Matematica, Università degli Studi di Milano, via Saldini 50, 20133 Milano, Italy

Dedicated to Jürgen Scheurle on the occasion of his retirement

Received  December 2017 Revised  April 2018 Published  April 2019

I give a short review of the theory of twisted symmetries of differential equations, emphasizing geometrical aspects. Some open problems are also mentioned.

Citation: Giuseppe Gaeta. On the geometry of twisted prolongations, and dynamical systems. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020070
References:
[1]

D. V. Alekseevsky, A. M. Vinogradov and V. V. Lychagin, Basic ideas and concepts of differential geometry, Geometry, I, 1–264, Encyclopaedia Math. Sci., 28, Springer, Berlin, 1991.

[2]

V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer, 1989. doi: 10.1007/978-1-4757-2063-1.

[3]

V. I. Arnold, Ordinary Differential Equations, Springer, 1992.

[4]

V. I. Arnold, Geometrical Methods in the Theory of Differential Equations, Springer, 1983.

[5]

A. Barco and G. E. Prince, Solvable symmetry structures in differential form, Acta Appl. Math., 66 (2001), 89-121. doi: 10.1023/A:1010609817442.

[6]

P. Basarab-Horwath, Integrability by quadratures for systems of involutive vector fields, Ukr. Math. J., 43 (1991), 1236-1242. doi: 10.1007/BF01061807.

[7]

J. F. CarinenaM. Del Olmo and P. Winternitz, On the relation between weak and strong invariance of differential equations, Lett. Math. Phys., 29 (1993), 151-163. doi: 10.1007/BF00749730.

[8]

D. Catalano Ferraioli, Nonlocal aspects of $\lambda$-symmetries and ODEs reduction, J. Phys. A: Math. Theor., 40 (2007), 5479-5489. doi: 10.1088/1751-8113/40/21/002.

[9]

D. Catalano Ferraioli and P. Morando, Local and nonlocal solvable structures in the reduction of ODEs, J. Phys. A: Math. Theor., 42 (2009), 035210 (15pp). doi: 10.1088/1751-8113/42/3/035210.

[10]

D. Catalano Ferraioli and P. Morando, Exploiting solvable structures in the integration of variational ordinary differential equations, preprint.

[11]

G. Cicogna, Reduction of systems of first-order differential equations via $\Lambda$-symmetries, Phys. Lett. A, 372 (2008), 3672-3677. doi: 10.1016/j.physleta.2008.02.041.

[12]

G. Cicogna, Symmetries of Hamiltonian equations and $\Lambda$-constants of motion, J. Nonlin. Math. Phys., 16 (2009), 43-60. doi: 10.1142/S1402925109000315.

[13]

G. Cicogna and G. Gaeta, Symmetry and Perturbation Theory in Nonlinear Dynamics, Springer, 1999.

[14]

G. Cicogna and G. Gaeta, Partial Lie-point symmetries of differential equations, J. Phys. A, 34 (2001), 491-512. doi: 10.1088/0305-4470/34/3/312.

[15]

G. Cicogna and G. Gaeta, Noether theorem for $\mu$-symmetries, J. Phys. A, 40 (2007), 11899-11921. doi: 10.1088/1751-8113/40/39/013.

[16]

G. CicognaG. Gaeta and P. Morando, On the relation between standard and $\mu$-symmetries for PDEs, J. Phys. A, 37 (2004), 9467-9486. doi: 10.1088/0305-4470/37/40/010.

[17]

G. Cicogna, G. Gaeta and S. Walcher, A generalization of $\lambda$-symmetry reduction for systems of ODEs: $\sigma$-symmetries, J. Phys. A, 45 (2012), 355205 (29pp). doi: 10.1088/1751-8113/45/35/355205.

[18]

G. CicognaG. Gaeta and S. Walcher, Orbital reducibility and a generalization of $\lambda$-symmetries, J. Lie Theory, 23 (2013), 357-381.

[19]

G. Cicogna, G. Gaeta and S. Walcher, Dynamical systems and $\sigma$-symmetries, J. Phys. A, 46 (2013), 235204 (23pp). doi: 10.1088/1751-8113/46/23/235204.

[20]

G. CicognaG. Gaeta and S. Walcher, Side conditions for ordinary differential equations, J. Lie Theory, 25 (2015), 125-146.

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G. Gaeta, Twisted symmetries of differential equations, J. Nonlin. Math. Phys., 16 (2009), S107–S136. doi: 10.1142/S1402925109000352.

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G. Gaeta, Simple and collective twisted symmetries, J. Nonlin. Math. Phys., 21 (2014), 593-627. doi: 10.1080/14029251.2014.975530.

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G. Gaeta, Smooth changes of frame and prolongations of vector fields, Int. J. Geom. Meths. Mod. Phys., 4 (2007), 807-827. doi: 10.1142/S0219887807002302.

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G. Gaeta, A gauge-theoretic description of $\mu$-prolongations, and $\mu$-symmetries of differential equations, J. Geom. Phys., 59 (2009), 519-539. doi: 10.1016/j.geomphys.2009.01.004.

[25]

G. Gaeta, Gauge fixing and twisted prolongations, J. Phys. A, 44 (2011), 325203 (9 pp). doi: 10.1088/1751-8113/44/32/325203.

[26]

G. Gaeta, Symmetry and Lie-Frobenius reduction of differential equations, J.Phys. A, 48 (2015), 015202, 22pp. doi: 10.1088/1751-8113/48/1/015202.

[27]

G. Gaeta, Symmetry of stochastic non-variational differential equations, Phys. Rep., 686 (2017), 1–62 [Erratum, Phys. Rep., 713 (2017), 18]. doi: 10.1016/j.physrep.2017.05.005.

[28]

G. Gaeta and G. Cicogna, Twisted symmetries and integrable systems, Int. J. Geom. Meths. Mod. Phys., 6 (2009), 1305-1321. doi: 10.1142/S0219887809004235.

[29]

G. GaetaF. D. GrosshansJ. Scheurle and S. Walcher, Reduction and reconstruction for symmetric ordinary differential equations, J. Diff. Eqs., 244 (2008), 1810-1839. doi: 10.1016/j.jde.2008.01.009.

[30]

G. Gaeta and P. Morando, On the geometry of lambda-symmetries and PDE reduction, J. Phys. A, 37 (2004), 6955-6975. doi: 10.1088/0305-4470/37/27/007.

[31]

G. Gaeta and S. Walcher, Dimension increase and splitting for Poincaré-Dulac normal forms, J. Nonlin. Math. Phys., 12 (2005), 327-342. doi: 10.2991/jnmp.2005.12.s1.26.

[32]

G. Gaeta and S. Walcher, Embedding and splitting ordinary differential equations in normal form, J. Diff. Eqs., 224 (2006), 98-119. doi: 10.1016/j.jde.2005.06.025.

[33]

C. Godbillon, Géométrie Différentielle et Mécanique Analitique, Hermann, 1969.

[34]

K. P. Hadeler and S. Walcher, Reducible ordinary differential equations, J. Nonlinear Sci., 16 (2006), 583-613. doi: 10.1007/s00332-004-0627-8.

[35]

T. Hartl and C. Athorne, Solvable structures and hidden symmetries, J. Phys. A: Math. Gen., 27 (1994), 3463-3474. doi: 10.1088/0305-4470/27/10/022.

[36]

Y. Kosmann-Schwarzbach, The Noether Theorems. Invariance and Conservation laws in the Twentieth Century, Springer, 2011. doi: 10.1007/978-0-387-87868-3.

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I. S. Krasil'schik and A. M. Vinogradov, Symmetries and Conservation Laws for Differential Equations of Mathematical Physics, A.M.S., 1999.

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D. LeviM. C. Nucci and M. A. Rodriguez, $\lambda$-symmetries for the reduction of continuous and discrete equations, Acta Appl. Math, 122 (2012), 311-321. doi: 10.1007/s10440-012-9745-8.

[39]

D. Levi and M. A. Rodriguez, $\lambda$-symmetries for discrete equations, J. Phys. A, 43 (2010), 292001, 9pp. doi: 10.1088/1751-8113/43/29/292001.

[40]

D. Levi and P. Winternitz, Non-classical symmetry reduction: Example of the Boussinesq equation, J. Phys. A, 22 (1989), 2915-2924. doi: 10.1088/0305-4470/22/15/010.

[41]

T. Levi Civita, Sulla determinazione di soluzioni particolari di un sistema canonico quando se ne conosce qualche integrale o relazione invariante, R. Accad. Lincei Ser. V, 10 (1901).

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P. Morando, Deformation of Lie derivative and $\mu$-symmetries, J. Phys. A, 40 (2007), 11547-11559. doi: 10.1088/1751-8113/40/38/007.

[43]

P. Morando, Reduction by $\lambda$-symmetries and $\sigma$-symmetries: A Frobenius approach, J. Nonlin. Math. Phys., 22 (2015), 47-59. doi: 10.1080/14029251.2015.996439.

[44]

C. Muriel and J. L. Romero, New methods of reduction for ordinary differential equations, IMA J. Appl. Math., 66 (2001), 111-125. doi: 10.1093/imamat/66.2.111.

[45]

C. Muriel and J. L. Romero, $C^\infty$ symmetries and nonsolvable symmetry algebras, IMA J. Appl. Math., 66 (2001), 477-498. doi: 10.1093/imamat/66.5.477.

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C. Muriel and J. L. Romero, Prolongations of vector fields and the invariants-by-derivation property, Theor. Math. Phys., 113 (2002), 1565-1575.

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C. Muriel and J. L. Romero, $C^\infty$-symmetries and integrability of ordinary differential equations, Proceedings of the I Colloquium on Lie Theory and Applications (Vigo), 35 (2002), 143-150.

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C. Muriel and J. L. Romero, $C^\infty$ symmetries and reduction of equations without Lie point symmetries, J. Lie Theory, 13 (2003), 167-188.

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C. Muriel and J. L. Romero, $C^\infty$-symmetries and nonlocal symmetries of exponential type, IMA J. Appl. Math., 72 (2007), 191-205. doi: 10.1093/imamat/hxm001.

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C. Muriel and J. L. Romero, Integrating factors and lambda-symmetries, J. Nonlin. Math. Phys., 15 (2008), 300-309. doi: 10.2991/jnmp.2008.15.s3.29.

[51]

C. Muriel and J. L. Romero, First integrals, integrating factors and $\lambda$-symmetries of second-order differential equations, J. Phys. A, 42 (2009), 365207, 17pp. doi: 10.1088/1751-8113/42/36/365207.

[52]

C. Muriel and J. L. Romero, A $\lambda$-symmetry-based method for the linearization and determination of first integrals of a family of second-order ordinary differential equations, J. Phys. A, 44 (2011), 245201, 19pp. doi: 10.1088/1751-8113/44/24/245201.

[53]

C. Muriel and J. L. Romero, Second-order differential equations with first integrals of the form $C (t)+ 1/(A (t, x) x + B (t, x))$, J. Nonlin. Math. Phys., 18 (2011), 237-250. doi: 10.1142/S1402925111001398.

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C. Muriel and J. L. Romero, Nonlocal symmetries, telescopic vector fields and $\lambda$-symmetries of ordinary differential equations, Symmetry, Integrability and Geometry: Methods and Applications, 8 (2012), Paper 106, 21 pp. doi: 10.3842/SIGMA.2012.106.

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C. Muriel and J. L. Romero, The $\lambda$-symmetry reduction method and Jacobi last multipliers, Comm. Nonlin. Science Num. Sim., 19 (2014), 807-820. doi: 10.1016/j.cnsns.2013.07.027.

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C. MurielJ. L. Romero and P. J. Olver, Variational $C^\infty$ symmetries and Euler-Lagrange equations, J. Diff. Eqs., 222 (2006), 164-184. doi: 10.1016/j.jde.2005.01.012.

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show all references

References:
[1]

D. V. Alekseevsky, A. M. Vinogradov and V. V. Lychagin, Basic ideas and concepts of differential geometry, Geometry, I, 1–264, Encyclopaedia Math. Sci., 28, Springer, Berlin, 1991.

[2]

V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer, 1989. doi: 10.1007/978-1-4757-2063-1.

[3]

V. I. Arnold, Ordinary Differential Equations, Springer, 1992.

[4]

V. I. Arnold, Geometrical Methods in the Theory of Differential Equations, Springer, 1983.

[5]

A. Barco and G. E. Prince, Solvable symmetry structures in differential form, Acta Appl. Math., 66 (2001), 89-121. doi: 10.1023/A:1010609817442.

[6]

P. Basarab-Horwath, Integrability by quadratures for systems of involutive vector fields, Ukr. Math. J., 43 (1991), 1236-1242. doi: 10.1007/BF01061807.

[7]

J. F. CarinenaM. Del Olmo and P. Winternitz, On the relation between weak and strong invariance of differential equations, Lett. Math. Phys., 29 (1993), 151-163. doi: 10.1007/BF00749730.

[8]

D. Catalano Ferraioli, Nonlocal aspects of $\lambda$-symmetries and ODEs reduction, J. Phys. A: Math. Theor., 40 (2007), 5479-5489. doi: 10.1088/1751-8113/40/21/002.

[9]

D. Catalano Ferraioli and P. Morando, Local and nonlocal solvable structures in the reduction of ODEs, J. Phys. A: Math. Theor., 42 (2009), 035210 (15pp). doi: 10.1088/1751-8113/42/3/035210.

[10]

D. Catalano Ferraioli and P. Morando, Exploiting solvable structures in the integration of variational ordinary differential equations, preprint.

[11]

G. Cicogna, Reduction of systems of first-order differential equations via $\Lambda$-symmetries, Phys. Lett. A, 372 (2008), 3672-3677. doi: 10.1016/j.physleta.2008.02.041.

[12]

G. Cicogna, Symmetries of Hamiltonian equations and $\Lambda$-constants of motion, J. Nonlin. Math. Phys., 16 (2009), 43-60. doi: 10.1142/S1402925109000315.

[13]

G. Cicogna and G. Gaeta, Symmetry and Perturbation Theory in Nonlinear Dynamics, Springer, 1999.

[14]

G. Cicogna and G. Gaeta, Partial Lie-point symmetries of differential equations, J. Phys. A, 34 (2001), 491-512. doi: 10.1088/0305-4470/34/3/312.

[15]

G. Cicogna and G. Gaeta, Noether theorem for $\mu$-symmetries, J. Phys. A, 40 (2007), 11899-11921. doi: 10.1088/1751-8113/40/39/013.

[16]

G. CicognaG. Gaeta and P. Morando, On the relation between standard and $\mu$-symmetries for PDEs, J. Phys. A, 37 (2004), 9467-9486. doi: 10.1088/0305-4470/37/40/010.

[17]

G. Cicogna, G. Gaeta and S. Walcher, A generalization of $\lambda$-symmetry reduction for systems of ODEs: $\sigma$-symmetries, J. Phys. A, 45 (2012), 355205 (29pp). doi: 10.1088/1751-8113/45/35/355205.

[18]

G. CicognaG. Gaeta and S. Walcher, Orbital reducibility and a generalization of $\lambda$-symmetries, J. Lie Theory, 23 (2013), 357-381.

[19]

G. Cicogna, G. Gaeta and S. Walcher, Dynamical systems and $\sigma$-symmetries, J. Phys. A, 46 (2013), 235204 (23pp). doi: 10.1088/1751-8113/46/23/235204.

[20]

G. CicognaG. Gaeta and S. Walcher, Side conditions for ordinary differential equations, J. Lie Theory, 25 (2015), 125-146.

[21]

G. Gaeta, Twisted symmetries of differential equations, J. Nonlin. Math. Phys., 16 (2009), S107–S136. doi: 10.1142/S1402925109000352.

[22]

G. Gaeta, Simple and collective twisted symmetries, J. Nonlin. Math. Phys., 21 (2014), 593-627. doi: 10.1080/14029251.2014.975530.

[23]

G. Gaeta, Smooth changes of frame and prolongations of vector fields, Int. J. Geom. Meths. Mod. Phys., 4 (2007), 807-827. doi: 10.1142/S0219887807002302.

[24]

G. Gaeta, A gauge-theoretic description of $\mu$-prolongations, and $\mu$-symmetries of differential equations, J. Geom. Phys., 59 (2009), 519-539. doi: 10.1016/j.geomphys.2009.01.004.

[25]

G. Gaeta, Gauge fixing and twisted prolongations, J. Phys. A, 44 (2011), 325203 (9 pp). doi: 10.1088/1751-8113/44/32/325203.

[26]

G. Gaeta, Symmetry and Lie-Frobenius reduction of differential equations, J.Phys. A, 48 (2015), 015202, 22pp. doi: 10.1088/1751-8113/48/1/015202.

[27]

G. Gaeta, Symmetry of stochastic non-variational differential equations, Phys. Rep., 686 (2017), 1–62 [Erratum, Phys. Rep., 713 (2017), 18]. doi: 10.1016/j.physrep.2017.05.005.

[28]

G. Gaeta and G. Cicogna, Twisted symmetries and integrable systems, Int. J. Geom. Meths. Mod. Phys., 6 (2009), 1305-1321. doi: 10.1142/S0219887809004235.

[29]

G. GaetaF. D. GrosshansJ. Scheurle and S. Walcher, Reduction and reconstruction for symmetric ordinary differential equations, J. Diff. Eqs., 244 (2008), 1810-1839. doi: 10.1016/j.jde.2008.01.009.

[30]

G. Gaeta and P. Morando, On the geometry of lambda-symmetries and PDE reduction, J. Phys. A, 37 (2004), 6955-6975. doi: 10.1088/0305-4470/37/27/007.

[31]

G. Gaeta and S. Walcher, Dimension increase and splitting for Poincaré-Dulac normal forms, J. Nonlin. Math. Phys., 12 (2005), 327-342. doi: 10.2991/jnmp.2005.12.s1.26.

[32]

G. Gaeta and S. Walcher, Embedding and splitting ordinary differential equations in normal form, J. Diff. Eqs., 224 (2006), 98-119. doi: 10.1016/j.jde.2005.06.025.

[33]

C. Godbillon, Géométrie Différentielle et Mécanique Analitique, Hermann, 1969.

[34]

K. P. Hadeler and S. Walcher, Reducible ordinary differential equations, J. Nonlinear Sci., 16 (2006), 583-613. doi: 10.1007/s00332-004-0627-8.

[35]

T. Hartl and C. Athorne, Solvable structures and hidden symmetries, J. Phys. A: Math. Gen., 27 (1994), 3463-3474. doi: 10.1088/0305-4470/27/10/022.

[36]

Y. Kosmann-Schwarzbach, The Noether Theorems. Invariance and Conservation laws in the Twentieth Century, Springer, 2011. doi: 10.1007/978-0-387-87868-3.

[37]

I. S. Krasil'schik and A. M. Vinogradov, Symmetries and Conservation Laws for Differential Equations of Mathematical Physics, A.M.S., 1999.

[38]

D. LeviM. C. Nucci and M. A. Rodriguez, $\lambda$-symmetries for the reduction of continuous and discrete equations, Acta Appl. Math, 122 (2012), 311-321. doi: 10.1007/s10440-012-9745-8.

[39]

D. Levi and M. A. Rodriguez, $\lambda$-symmetries for discrete equations, J. Phys. A, 43 (2010), 292001, 9pp. doi: 10.1088/1751-8113/43/29/292001.

[40]

D. Levi and P. Winternitz, Non-classical symmetry reduction: Example of the Boussinesq equation, J. Phys. A, 22 (1989), 2915-2924. doi: 10.1088/0305-4470/22/15/010.

[41]

T. Levi Civita, Sulla determinazione di soluzioni particolari di un sistema canonico quando se ne conosce qualche integrale o relazione invariante, R. Accad. Lincei Ser. V, 10 (1901).

[42]

P. Morando, Deformation of Lie derivative and $\mu$-symmetries, J. Phys. A, 40 (2007), 11547-11559. doi: 10.1088/1751-8113/40/38/007.

[43]

P. Morando, Reduction by $\lambda$-symmetries and $\sigma$-symmetries: A Frobenius approach, J. Nonlin. Math. Phys., 22 (2015), 47-59. doi: 10.1080/14029251.2015.996439.

[44]

C. Muriel and J. L. Romero, New methods of reduction for ordinary differential equations, IMA J. Appl. Math., 66 (2001), 111-125. doi: 10.1093/imamat/66.2.111.

[45]

C. Muriel and J. L. Romero, $C^\infty$ symmetries and nonsolvable symmetry algebras, IMA J. Appl. Math., 66 (2001), 477-498. doi: 10.1093/imamat/66.5.477.

[46]

C. Muriel and J. L. Romero, Prolongations of vector fields and the invariants-by-derivation property, Theor. Math. Phys., 113 (2002), 1565-1575.

[47]

C. Muriel and J. L. Romero, $C^\infty$-symmetries and integrability of ordinary differential equations, Proceedings of the I Colloquium on Lie Theory and Applications (Vigo), 35 (2002), 143-150.

[48]

C. Muriel and J. L. Romero, $C^\infty$ symmetries and reduction of equations without Lie point symmetries, J. Lie Theory, 13 (2003), 167-188.

[49]

C. Muriel and J. L. Romero, $C^\infty$-symmetries and nonlocal symmetries of exponential type, IMA J. Appl. Math., 72 (2007), 191-205. doi: 10.1093/imamat/hxm001.

[50]

C. Muriel and J. L. Romero, Integrating factors and lambda-symmetries, J. Nonlin. Math. Phys., 15 (2008), 300-309. doi: 10.2991/jnmp.2008.15.s3.29.

[51]

C. Muriel and J. L. Romero, First integrals, integrating factors and $\lambda$-symmetries of second-order differential equations, J. Phys. A, 42 (2009), 365207, 17pp. doi: 10.1088/1751-8113/42/36/365207.

[52]

C. Muriel and J. L. Romero, A $\lambda$-symmetry-based method for the linearization and determination of first integrals of a family of second-order ordinary differential equations, J. Phys. A, 44 (2011), 245201, 19pp. doi: 10.1088/1751-8113/44/24/245201.

[53]

C. Muriel and J. L. Romero, Second-order differential equations with first integrals of the form $C (t)+ 1/(A (t, x) x + B (t, x))$, J. Nonlin. Math. Phys., 18 (2011), 237-250. doi: 10.1142/S1402925111001398.

[54]

C. Muriel and J. L. Romero, Nonlocal symmetries, telescopic vector fields and $\lambda$-symmetries of ordinary differential equations, Symmetry, Integrability and Geometry: Methods and Applications, 8 (2012), Paper 106, 21 pp. doi: 10.3842/SIGMA.2012.106.

[55]

C. Muriel and J. L. Romero, The $\lambda$-symmetry reduction method and Jacobi last multipliers, Comm. Nonlin. Science Num. Sim., 19 (2014), 807-820. doi: 10.1016/j.cnsns.2013.07.027.

[56]

C. MurielJ. L. Romero and P. J. Olver, Variational $C^\infty$ symmetries and Euler-Lagrange equations, J. Diff. Eqs., 222 (2006), 164-184. doi: 10.1016/j.jde.2005.01.012.

[57]

P. J. Olver, Application of Lie Groups to Differential Equations, Springer, 1986. doi: 10.1007/978-1-4684-0274-2.

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