An algorithm for determining polynomial first integrals of autonomous systems of ordinary differential equations (Q1070778)

The problem under discussion here is the determination of a set of functions f(x), where \(x=(x_ 1,...,x_ n)\), having the property that their total first derivatives with respect to time, t, vanish when x satisfies a system of n autonomous differential equations of first order. The functions f may not depend on t explicitly, or may involve an exponential factor including t. The system of differential equations usually involves only polynomials, together with one or more parameters to be determined. It appears that a knowledge of these first integrals f is of great assistance in characterising the solution sets of the differential equations. The author gives a general outline of a package for solving problems of this nature when the time-independent parts of the functions f are assumed to be of polynomial form. In essence a very large number of algebraic equations is solved, taking account of the special structure originating from the problem. A rather simple illustration of this process is afforded by the Lorenz equations, for which the author lists the first integrals up to and including order 6.