Explicit Construction of First Integrals with Quasi-monomial Terms from the Painlev\'{e} Series

The Painlev'{e} and weak Painlev'{e} conjectures have been used widely to identify new integrable nonlinear dynamical systems. For a system which passes the Painlev'{e} test, the calculation of the integrals relies on a variety of methods which are independent from Painlev'{e} analysis. The present paper proposes an explicit algorithm to build first integrals of a dynamical system, expressed as `quasi-polynomial’ functions, from the information provided solely by the Painlev'{e} - Laurent series solutions of a system of ODEs. Restrictions on the number and form of quasi-monomial terms appearing in a quasi-polynomial integral are obtained by an application of a theorem by Yoshida (1983). The integrals are obtained by a proper balancing of the coefficients in a quasi-polynomial function selected as initial ansatz for the integral, so that all dependence on powers of the time $\tau =t-t_0$ is eliminated. Both right and left Painlev'{e} series are useful in the method. Alternatively, the method can be used to show the non-existence of a quasi-polynomial first integral. Examples from specific dynamical systems are given.