Corrections and complements to “Liouvillian integration of the Lotka-Volterra system”

Abstract

The Lotka-Volterra system of autonomous differential equations consists in three homogeneous polynomial equations of degree 2 in three variables.

This system, or the corresponding vector fieldLV(A,B,C), depends on three non-zero (complex) parameters and may be written as

$$LV(A,B,C) = x(Cy + z)\partial _x + y(Az + x)\partial _y + z(Bx + y)\partial _z $$

.

In fact,LV(A,B,C) can be chosen as a normal form for most of the factored quadratic systems; the study of its first integrals of degree 0 is thus of great mathematical interest.

In the paper into consideration [1], we thus described all possible values of the triple (A,B,C) of non-zero parameters for whichLV(A,B,C) has a homogeneous liouvillian first integral of degree 0.

We also discussed the corresponding problem of the liouvillian integration for quadratic factored vector fields that cannot be put in Lotka-Volterra normal form, for instance with some 0 amongA,B,C,.

There are some errors in the description of these marginal situations that we would like to correct in the present note.