ИСТИНА

To study the polynomial modification of the Li\'enard equation with a 4th-order nonlinearity, a system of algebraic equations is constructed with respect to the parameters of the system. The sets of parameters satisfying it correspond to cases of integrability. However, it is not possible to find rational solutions of this algebraic system in general form. In this case, we know one set of numerical values of the parameters for which integrability takes place. This set satisfies the given algebraic system also, i.e. is its particular solution. Using the form of this known solution, we can reduce the number of independent parameters so that it becomes possible to find two rational solutions. One solution is trivial, the second allows us to find a two-dimensional manifold of parameter values for which the given ODE system has a first integral.