In this article we prove the following result: if two 2-dimensional 2-homogeneous rational vector fields commute, then either both vector fields can be explicitly integrated to produce rational flows with orbits being lines through the origin, or both flows can be explicitly integrated in terms of algebraic functions. In the latter case, orbits of each flow are given in terms of 1-homogeneous rational functions W as curves W(x,y)=const. An exhaustive method to construct such commuting algebraic flows is presented. The degree of the so-obtained algebraic functions in two variables can be arbitrarily high.
Submitted March 23, 2018. Published July 3, 2018.
Math Subject Classifications: 34A30, 37C10, 14H05, 35F05, 14E07.
Key Words: Translation equation; flow; rational vector fields; linear ODEs; autonomous non-linear ODEs; first order linear PDEs; algebraic functions; Lie bracket; commuting flows; Cremona groups; Wronskian.
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| Giedrius Alkauskas |
Institute of Computer Science
Naugarduko 24, LT-03225 Vilnius, Lithuania
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