  We prove that any piece of a rotational hypersurface with prescribed mean
curvature function in a Euclidean space can be uniquely extended infinitely,
which generalizes the results by Euler and Delaunay for surfaces of revolution
with constant mean curvautre. Next, we prove the same kind of theorem for
generalized rotational hypersurfaces of O(l+1) x O(m+1)-type. The key lemmas in
this paper show the existence of solutions for singular initial value problems
which arise from the analysis of ordinary differential equations of generating
curves of those hypersurfaces.
