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 field $\lv$, depends on three non-zero
(complex) parameters and may be written as
$\lv=V_x\partial_x+V_y\partial_y+V_z\partial_z$ where \[V_x=x ( C y +
z),\;V_y=y ( A z + x),\;V_z=z ( B x + y).\] Similar systems of equations have
been studied by Volterra in his mathematical approach of the competition of
species and this is the reason why this name has been given to such systems.
In fact, $\lv$ can be chosen as a normal form for most of factored
quadratic systems; the study of its first integrals of degree 0 is thus of
great mathematical interest.
Given a homogeneous vector field, there is a foliation whose leaves are
homogeneous surfaces in the three-dimensional space (or curves in the
corresponding projective plane), such that the trajectories of the vector field
are completely contained in a leaf. A first integral of degree 0 is then a
function on the set of all leaves of the previous foliation.
In the present paper, we give all values of the triple $(A,B,C)$ of non-zero
parameters for which $\lv$ has a homogeneous liouvillian first integral of
degree $0$. We also discuss the corresponding problem of liouvillian
integration for quadratic factored vector fields that cannot be put in
Lotka-Volterra normal form.
Our proof essentially relies on combinatorics and elementary algebraic
geometry, especially in proving that some conditions are necessary.