  In this paper, we consider a translating soliton for the mean curvature flow
starting from a graph of a function on a domain in a unit sphere which is
constant along each leaf of isoparametric foliation. First, we show that such a
function is given as a composition of an isoparametric function on the sphere
and a function which is given as a solution of a certain ordinary differential
equation. Further, we analyze the shape of the graphs of the solutions of the
ordinary differential equation. This analysis leads to the classification of
the shape of such translating solitons. Finally, we investigate a domain of the
function which is given as a composition of the isoparametric function and the
solution of the ordinary differential equation in the case where the number of
distinct principal curvatures of the isoparametric hypersurface defined by the
regular level set for the isoparametric function is 1, 2, or 3.
