Minimal surfaces in Euclidean space with a log-linear density

We study surfaces in Euclidean space ${\mathbb R}^3$ that are minimal for a log-linear density $\phi(x,y,z)=\alpha x+\beta y+\gamma y$, where $\alpha,\beta,\gamma$ are real numbers not all zero. We prove that if a surface is $\phi$-minimal foliated by circles in parallel planes, then these planes are orthogonal to the vector $(\alpha,\beta,\gamma)$ and the surface must be rotational. We also classify all minimal surfaces of translation type.