Global Properties of Kovalevskaya Exponents

This paper contains a collection of properties of Kovalevskaya exponents which are eigenvalues of a linearization matrix of weighted homogeneous nonlinear systems along certain straight-line particular solutions. Relations in the form of linear combinations of Kovalevskaya exponents with nonnegative integers related to the presence of first integrals of the weighted homogeneous nonlinear systems have been known for a long time. As a new result other nonlinear relations between Kovalevskaya exponents calculated on all straight-line particular solutions are presented. They were obtained by an application of the Euler–Jacobi–Kronecker formula specified to an appropriate n-form in a certain weighted homogeneous projective space.