On the integrability of some three-dimensional Lotka-Volterra equations with rank-$1$ resonances

Abstract

We investigate the local integrability in $\mathbb{C}^3$ of some three-dimensional Lotka-Volterra equations at the origin with $(p:q:r)$-resonance,

dot x &= P = x(p +ax+by+cz),

dot y &= Q = y(q + dx +ey + fz),

dot z &= R = z(r + gx + hy +kz).

Recent work on this problem has centered on the case where the resonance is of "rank\guio{$2$}". That is, there are two independent linear dependencies of $p$, $q$ and $r$ over~$\mathbb{Q}$. Here, we consider some situations where there is only one such dependency. In particular, we give necessary and sufficient conditions for integrability for the case of $(i,-i,\lambda)$-resonance with $\lambda\notin i\mathbb{R}$ (after a scaling, this is just the case $p+q=0$ with $q/r \notin \mathbb{R}$), and also the case of $(i-1,-i-1,2)$-resonance (a subcase of $p+q+r=0$) under the additional assumption that $a=k=0$.

Our necessary and sufficient conditions for integrability are given via the search for two independent first integrals of the form $x^\alpha y^\beta z^\gamma (1+O(x,y,z))$. However, a new feature in the case of rank-$1$ resonance is that there is a distinguished choice of analytic first integral, and hence it makes sense to seek conditions for just one (analytic) first integral to exist. We give necessary and sufficient conditions for just one first integral for the two families of systems mentioned above, except that for the second family some of the cases of sufficiency have been left as conjectural.