Abstract: I will present a (non-trivial) semi-algorithm that takes an irreducible polynomial p(x,y) and a rational function r(x,y) as input and outputs (a description of) all solutions of r(x,y) + q(x,y) p(x,y) = f(x) + g(y) where q(x,y) is a rational function whose denominator is not divisible by p(x,y) and f(x) and g(y) are univariate rational functions. The semi-algorithm presented does not necessarily terminate. It does if the equation has a (non-trivial) solution. However, if there is none, it might not terminate. Termination depends on a dynamical system on the curve defined by p(x,y) and the location of the poles of r(x,y). I will furthermore provide some context and explain how the above problem relates to the classification of generating functions of lattice walks restricted to cones. Further applications are: the computation of intersections of fields, which again has applications in computer vision, parameter identifiability of ODE models and algebraic (in)dependence of solutions of differential equations.