Geodesic
Painlev\'e test for ordinary differential equations associated with the heat equation
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Algebraic topology, convex polytopes, and related topics, Tome 286 (2014), pp. 75-87

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We consider nonlinear ordinary differential equations up to the sixth order that are associated with the heat equation. Each of them is subjected to the Painlevé analysis. For the fourth- and sixth-order equations we obtain a criterion for having the Painlevé property; for the fifth-order equation we formulate necessary conditions for passing the Painlevé test. We also present a fifth-order equation analogous to the Chazy-$3$ equation. 
@article{TM_2014_286_a4,
     author = {A. V. Vinogradov},
     title = {Painlev\'e test for ordinary differential equations associated with the heat equation},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {75--87},
     publisher = {mathdoc},
     volume = {286},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2014_286_a4/}
}
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A. V. Vinogradov. Painlev\'e test for ordinary differential equations associated with the heat equation. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Algebraic topology, convex polytopes, and related topics, Tome 286 (2014), pp. 75-87. http://geodesic.mathdoc.fr/item/TM_2014_286_a4/