  The aim of this paper is to investigate the mean curvature flow soliton
solutions on the Heisenberg group $\mathcal{H}$ when the initial data is a
ruled surface by straight lines. We give a family of those solutions which are
generated by $\mathfrak{Iso}_{0}(\mathcal{H})$ (the isometries of $\mathcal{H}$
for which the origin is a fix point). We conclude that the function which
describe the motion of these surfaces under MCF, is always a linear affine
function. As an application we proof that the Grim Reaper solution evolves from
a ruled surface in $\mathcal{H}$. We also provide other examples.
