Darboux polynomials, ecstatic curves and Kovalevskaya exponents

In 1878, Darboux put forward a novel approach for identifying invariants of systems of autonomous ordinary differential equations. This approach was subsequently elaborated upon by Poincaré, Penléve and Autonne in 1891. In the works of M.N.Lagutinskii in 1911–1912, two methods for constructing Darboux polynomials were proposed. These methods are now referred to as the Lagutinskii–Pereira algorithm and Lagutinskii–Levelt exponents in the international literature. The initial method permits generalizations, enabling the computation of rational first integrals and first integrals within the class of $k$-Darboux functions and Liouville functions. The second of the methods is based on the use of Kovalevskaya–Penlevé singular analysis methods to find the cofactors of Darboux polynomials. The talk discusses the Lagutinskii results and modern methods of constructing first integrals based on his ideas. As an example, we consider finding Darboux polynomials for full Toda lattice using modern computer technologies.