Separating Variables in Bivariate Polynomial Ideals: the Local Case

We present a semi-algorithm which for any irreducible $p\in \mathbb{K}[x,y]$ finds all elements of $\mathbb{K}(x)+\mathbb{K}(y)$ that are of the form $qp$ for some $q\in \mathbb{K}(x,y)$ whose denominator is not divisible by $p$.