  A soliton of the mean curvature flow in the product space
$\mathbb{s}^2\times\mathbb{R}$ as a surface whose mean curvature $H$ satisfies
the equation $H=\langle N,X\rangle$, where $N$ is the unit normal of the
surface and $X$ is a Killing vector field. In this paper we consider the vector
field tangent to the fibers and the vector field associated to a rotations
about an axis of $\mathbb{s}^2$, respectively. We give a classification of the
solitons with respect to these vector fields assuming that the surface is
invariant under a one-parameter group of vertical translations or under a group
of rotations of $\mathbb{s}^2$.
