Separated Variables on Plane Algebraic Curves

We investigate the problem of deciding whether the restriction of a rational function $r\in \mathbb{K}(x,y)$ to the curve associated with an irreducible polynomial $p\in \mathbb{K}[x,y]$ is the restriction of an element of $\mathbb{K}(x)+\mathbb{K}(y)$. We present an algorithm and a conjectural semi-algorithm for finding such elements depending on whether $p$ has a non-trivial rational multiple in $\mathbb{K}(x)+\mathbb{K}(y)$ or not.