Kovalevskaya exponents of the Riccati hierarchy

Abstract

We carry out a Kovalevskaya analysis of the Riccati hierarchy. We determine all indicial loci and Kovalevskaya exponents and identify a rigid recursive structure governing how free parameters enter Laurent solutions. We further identify a nontrivial quasi--homogeneous vector field commuting with the hierarchy and use it to obtain an explicit parametrization of all solutions in terms of a single polynomial. Notably, in the two-dimensional case, the same general solution is recovered by the blow-up resolution. Within this parametrization, collisions of poles correspond to degeneration limits of the principal Laurent family, through which lower indicial loci appear. Finally, negative Kovalevskaya exponents are interpreted analytically through annular Laurent expansions, which describe how different collections of poles dominate in different complex regions.