  Calabi observed that there is a natural correspondence between the solutions
of the minimal surface equation in $\mathbb{R}^3$ with those of the maximal
spacelike surface equation in $\mathbb{L}^3$. We are going to show how this
correspondence can be extended to the family of $\varphi $-minimal graphs in
$\mathbb{R}^3 $ when the function $\varphi$ is invariant under a two-parametric
group of translations. We give also applications in the study and description
of new examples.
