Darboux integrability for 3D Lotka-Volterra systems

We describe the improved Darboux theory of integrability for polynomial ord inary differential equations in three dimensions. Using this theory and com puter algebra, we study the existence of first integrals for the three-dime nsional Lotka-Volterra systems. Only working up to degree two with the inva riant algebraic surfaces and the exponential factors, we find the major par t of the known first integrals for such systems, and in addition we find th ree new classes of integrability. The method used is of general interest an d can be applied to any polynomial ordinary differential equations in arbit rary dimension.