Translation invariant surfaces in the 3-dimensional Heisenberg‎ ‎group

‎In this paper‎, ‎we study translation invariant surfaces in the‎ ‎3-dimensional Heisenberg group $\rm Nil_3$‎. ‎In particular‎, ‎we‎ ‎completely classify translation invariant surfaces in $\rm Nil_3$‎ ‎whose position vector $x$ satisfies the equation $\Delta x = Ax$‎, ‎where $\Delta$ is the Laplacian operator of the surface and $A$‎ ‎is a $3 \times 3$-real matrix‎.