Initial value problems on cohomogeneity one manifolds, I

We study initial value problems for various geometric equations on a cohomogeneity manifold near a singular orbit. We show that when prescribing the Ricci curvature, or finding solutions to the Einstein and soliton equations, there exist solutions near the singular orbit, unique up to a finite number of constants. In part I we make a special assumption that significantly simplifies the proof, and will solve the general case in Part II.