  In this paper we study $\varphi$-minimal surfaces in $\mathbb{R}^3$ when the
function $\varphi$ is invariant under a two-parametric group of translations.
Particularly those which are complete graphs over domains in $\mathbb{R}^2$. We
describe a full classification of complete flat embedded $\varphi$-minimal
surfaces if $\varphi$ is strictly monotone and characterize rotational
$\varphi$-minimal surfaces by its behavior at infinity when $\varphi$ has a
quadratic growth.
