Singularity methods for meromorphic solutions of differential equations

Abstract

Consider an N -th order algebraic ordinary differential equation (ODE) for u(x) which may or may not possess the Painleve´ property (PP), defined as the absence of movable critical singularities in the general solution, a singularity being called “critical” if multivaluedness takes place around it. This does not prevent the existence of particular solutions obeying a lower order ODE with the PP. Let us therefore address the problem to find all such solutions (i.e. without movable critical singularities) in closed form. We restrict here to autonomous ODEs, i.e. which do not depend explicitly on the independent variable x ∈ C.