Abstract
A method for finding exact solutions and the first integrals is presented. The basic idea of the method is to use the value of the Fuchs index that appears in the Painlevé test to construct the auxiliary equation for finding the first integrals and exact solutions of nonlinear differential equations. It allows us to obtain the first integrals and new exact solutions of some nonlinear ordinary differential equations. The main feature of the method is that we do not assign a solution function at the beginning, we find this function during calculations. This approach is conceptually equivalent to the third step of the Painlevé test and sometimes allows us to change this step. Our approach generalizes a number of other methods for finding exact solutions of nonlinear differential equations. We demonstrate a method for finding the traveling wave solutions and the first integrals of the well-known nonlinear evolution equation for description of surface waves in a convecting liquid. The general solution of this equation at some conditions on parameters and new traveling wave solutions of the fourth-order equation are found.
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