Higher order Briot—Bouquet differential equations
In this note k is supposed to be >2. The main problem of integration is to find necessary and, if possible, sufficient conditions for the existence of what here will be called canonical solutions of (1.2), that is, solutions which are transcendental single-valued functions, holomorphic save for poles in the finite plane. A more general problem, not considered here, is to find the equations with fixed critical points (branch points and essential singularities). It does not seem to be significant for BB equations. Various facts are known about this problem, a few valid for all k, more for k = 1 or 2. Those which are needed or suggestive for the investigation are stated as lemmata.