Rational Solutions of Riccati-like Partial Differential Equations

When factoring linear partial differential systems with a finite-dimensional solution space or analysing symmetries of nonlinear ODEs, we need to look for rational solutions of certain nonlinear PDEs. The nonlinear PDEs are called Riccati-like because they arise in a similar way as Riccati ODEs. In this paper we describe the structure of rational solutions of a Riccati-like system, and an algorithm for computing them. The algorithm is also applicable to finding all rational solutions of Lie’s system { ∂xu + u2 + a1u + a2v + a3, ∂yu + uv + b1u + b2v + b3, ∂xv + uv + c1u + c2v + c3, ∂yv + v2 + d1u + d2v + d3},where a1, . . . , d3are rational functions of x and y.