  We carry out a Painlev\'e analysis to find the cases where the cohomogeneity
one steady Ricci soliton equation can be integrable. We concentrate on two
classes of solitons: warped products and complex line bundles over a Fano
K\"ahler Einstein base. For warped products, the analysis singles out the case
with one factor where the dimension of the hypersurface is a perfect square,
with the $n=4$ particularly distinguished. The case with two factors each of
dimension $2$ is also singled out by the analysis. In the case of complex line
bundles, a 1-parameter family is singled out for every even dimension.
