  We establish the existence of real pole-free solutions to all even members of
the Painlev\'e I hierarchy. We also obtain asymptotics for those solutions and
describe their relevance in the description of critical asymptotic behavior of
solutions to the KdV equation in the small dispersion limit. This was
understood in the case of a generic critical point, and we generalize it here
to the case of non-generic critical points.
