Meromorphic functions in a first-order system

Abstract

We consider the differential equations $s' = c^5$ and $c' = - s^5$ with initial conditions $s(0) = 0$ and $c(0) = 1$. We show that $s^2 c^2$ extends to the reciprocal of the Weierstrass elliptic function with invariants $g_2 = 0$ and $g_3 = 16$. We also show that powers of $s^2 c^2$ are the only functions of the form $s^m c^n$ that extend meromorphically in the plane.