Meromorphic solutions of algebraic differential equations

where F is a polynomial in the first k+ 1 variables, whose coefficients are analytic functions of the independent variable z. If the conditions of Cauchy's theorem for the existence and uniqueness of the solution are satisfied, then (0.1) determines an analytic function in a neighbourhood of a given point z0. One of the most difficult problems in the analytic theory of differential equations is that of the analytic continuation of the solution and of studying it in the whole domain where it exists. It is natural, first of all, to ask whether there are solutions of (0.1) that are meromorphic in the finite plane C. A second important problem is to study the properties of meromorphic solutions immediately from the equation, if it is known that such solutions exist. This article is devoted to this second problem. Throughout what follows, unless stated otherwise, by a meromorphic function we mean one that is meromorphic in C. We always use ζ to denote an independent variable, ζ G C.