Electron. J. Differential Equations,
Vol. 2018 (2018), No. 138, pp. 121.
Planar 2homogeneous commutative rational vector fields
Giedrius Alkauskas
Abstract:
In this article we prove the following result:
if two 2dimensional 2homogeneous 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 1homogeneous
rational functions W as curves W(x,y)=const.
An exhaustive method to construct such commuting algebraic flows is presented.
The degree of the soobtained 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 nonlinear ODEs; first order linear PDEs; algebraic functions;
Lie bracket; commuting flows; Cremona groups; Wronskian.
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Giedrius Alkauskas
Vilnius University
Institute of Computer Science
Naugarduko 24, LT03225 Vilnius, Lithuania
email: giedrius.alkauskas@mif.vu.lt

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