Meromorphic solutions of autonomous ordinary differential equations without the finiteness property
We consider the problem of finding meromorphic solutions of autonomous algebraic ordinary differential equations possessing an infinite number of local solutions given by Laurent series in neighborhoods of poles. We present an algebraic-geometric method for finding all meromorphic solutions that are elliptic or rational in the exponential function. As an example, we classify such families of meromorphic solutions for fourth-order ordinary differential equations arising via integrating once traveling-wave reductions of the generalized Rosenau–Korteweg–de Vries equation and the fifth-order Korteweg–de Vries–Burgers equation. Solutions important from a physical point of view are presented.