Integrals of quadratic ordinary differential equations in R3: The Lotka-Volterra system

A method already introduced by the last two authors for finding the integrable cases of three-dimensional autonomous ordinary differential equations based on the Frobenius integrability theorem is described in detail. Using this method and computer algebra, the so-called three-dimensional Lotka-Volterra system is studied. Many cases of integrability are thus found. The study of this system is completed by the application of Painlevé analysis and the Jacobi last multiplier method. The methods used are of general interest and can be applied to many other systems.