Geodesic
Hamiltonians associated with the~third and fifth Painlev\'e equations
Teoretičeskaâ i matematičeskaâ fizika, Tome 162 (2010) no. 1, pp. 69-74

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We obtain a Painlevé-type differential equation for the simplest rational Hamiltonian associated with the fifth Painlevé equation in the case $\gamma\ne0$, $\delta=0$. We prove the existence of Hamiltonians of a nonrational type associated with the fifth Painlevé equation in the case $\gamma\ne0$, $\delta=0$. We obtain a generalization of the Garnier and Okamoto formulas for rational Hamiltonians associated with the third Painlevé equation.
Keywords: third Painlevé equation, fifth Painlevé equation, Hamiltonian. 
@article{TMF_2010_162_1_a2,
     author = {V. V. Tsegel'nik},
     title = {Hamiltonians associated with the~third and fifth {Painlev\'e} equations},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {69--74},
     publisher = {mathdoc},
     volume = {162},
     number = {1},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2010_162_1_a2/}
}
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V. V. Tsegel'nik. Hamiltonians associated with the~third and fifth Painlev\'e equations. Teoretičeskaâ i matematičeskaâ fizika, Tome 162 (2010) no. 1, pp. 69-74. http://geodesic.mathdoc.fr/item/TMF_2010_162_1_a2/