Kovalevskaya exponents of the Riccati hierarchy

TL;DR

This paper performs a Kovalevskaya analysis of the Riccati hierarchy, revealing its recursive structure, explicit solutions, and pole collision behaviors, advancing understanding of its integrability properties.

Contribution

It identifies the indicial loci, Kovalevskaya exponents, and a recursive structure, providing explicit parametrizations and analytical interpretations of solutions.

Findings

Determined all indicial loci and Kovalevskaya exponents.

Identified a recursive structure governing free parameters.

Provided explicit solution parametrization and pole collision analysis.

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.