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 field LV(A,B,C), depends on three non-zero (complex) parameters and may be written as LV(A,B,C)=Vx∂x+Vy∂y+Vz∂z whereimageVx=x(Cy+z), Vy=y(Az+x), Vz=z(Bx+y).As LV(A,B,C) is homogeneous, there is a foliation whose leaves are homogeneous surfaces in the three-dimensional space image, or curves in the corresponding projective plane image, such that the trajectories of the vector field are completely contained in a leaf. An homogeneous first integral of degree 0 is then a non-constant function on the set of all leaves of the previous foliation.
