Suppose $(A\otimes 1 + C\otimes B)x\otimes y = 0$, then $y$ is an eigen-vector of $B$ with some eigen-value $\lambda$ and $(A + \lambda C)x = 0$. The Laplacian in spherical coordinates is an operator of this type!

We assume that $x\neq 0$ and $y\neq 0$. We have $$ 0 = (A\otimes 1 + C\otimes B)x\otimes y = A\otimes 1(x\otimes y) + C\otimes B(x\otimes y) = Ax\otimes y + Cx\otimes By. $$ This implies that $Ax\otimes y = (-Cx)\otimes By$. By Exam, we conclude that $Ax = -\lambda Cx$ and $y = \lambda^{-1}By$ for some $\lambda$ ($\neq 0$). Thus, $(A + \lambda C)x = 0$ and $By = \lambda y$.