Integration of a Degenerate System of ODEs

The integrability of a two-dimensional autonomous polynomial system of ordinary differential equations (ODEs) with a degenerate singular point at the origin that depends on six parameters is investigated. The integrability condition for the first quasihomogeneous approximation allows one of these parameters to be fixed on a countable set of values. The further analysis is carried out for this value and five free parameters. Using the power geometry method, the system is reduced to a non-degenerate form through the blowup process. Then, the necessary conditions for its local integrability are calculated using the method of normal forms. In other words, the conditions for the parameters under which the original system is locally integrable near the degenerate stationary point are found. By resolving these conditions, we find seven two-parameter families in the five-dimensional parametric space. For parameter values from these families, the first integrals of the system are found. The cumbersome calculations that occur in the problem under consideration are carried out using computer algebra.