Some generalizations of Halphen's equations

Abstract

Halphen's equations are given by a remarkable polynomial vector field in $\mathbf{C}^{3}$ having only single-valued solutions, defined in domains bounded by a circle or by a line. By generalizing the Lie-theoretic principle behind Halphen's equations and borrowing some facts from the theory of deformations of Fuchsian groups, we exhibit a family of polynomial vector fields in $\mathbf{C}^{3}$ having only single-valued solutions. The solutions of vector fields within this family are defined in domains which had not been previously observed as domains of definition of solutions of polynomial vector fields in $\mathbf{C}^{3}$. For example, we obtain polynomial vector fields having solutions defined in domains that are bounded by a fractal curve.