Geodesic
On the Maupertuis--Jakobi transformation and Toda chains
Zapiski Nauchnykh Seminarov POMI, Questions of quantum field theory and statistical physics. Part 16, Tome 269 (2000), pp. 354-364

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For the Toda lattice we consider properties of canonical transformations of the extended phase space, which preserve integrability. The separated variables are invariant under change of the time. On the other hand, mapping of the time induces transformations of the action-angles variables and a shift of the generating function of the Bäcklund transformation. 
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     author = {A. V. Tsiganov},
     title = {On the {Maupertuis--Jakobi} transformation and {Toda} chains},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {354--364},
     publisher = {mathdoc},
     volume = {269},
     year = {2000},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2000_269_a23/}
}
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A. V. Tsiganov. On the Maupertuis--Jakobi transformation and Toda chains. Zapiski Nauchnykh Seminarov POMI, Questions of quantum field theory and statistical physics. Part 16, Tome 269 (2000), pp. 354-364. http://geodesic.mathdoc.fr/item/ZNSL_2000_269_a23/