Electronic Journal of Differential Equations,
Vol. 2015 (2015), No. 310, pp. 1-7.

Title: Fields of rational constants of cyclic factorizable derivations

Authors: Janusz Zielinski (N. Copernicus Univ., Torun, Poland)

Abstract: 
 We describe all rational constants of a large family of four-variable
 cyclic factorizable derivations. Thus, we determine all rational
 first integrals of their corresponding systems of differential equations.
 Moreover, we give a characteristic of all four-variable Lotka-Volterra 
 derivations  with a nontrivial rational constant.
 All considerations are over an arbitrary field of characteristic zero.
 Our main tool is the investigation of the cofactors of strict Darboux 
 polynomials.  Factorizable derivations are important in derivation theory.
 Namely, we may associate the factorizable  derivation with any given
 derivation of a polynomial ring and that construction
 helps to determine rational constants of arbitrary derivations.
 Besides, Lotka-Volterra systems play a significant role in population 
 biology, laser physics and plasma physics.

Submitted November 12, 2014. Published December 21, 2015.
Math Subject Classifications: 34A34, 13N15, 12H05, 92D25.
Key Words: Lotka-Volterra derivation; factorizable derivation;
           rational constant; rational first integral.