We prove that for a given continuous function $H(s)$, $(-\infty$ < $s$ < $\infty)$, there exists a globally defined generating curve of a rotational hypersurface in a Euclidean space such that the mean curvature is $H(s)$. We also prove a similar theorem for generalized rotational hypersurfaces of $O(l+1)\times O(m+1)$-type. The key lemmas in this paper show the existence of solutions for singular initial value problems of ordinary differential equations satisfied using generating curves of those hypersurfaces.
"Global existence of generalized rotational hypersurfaces with prescribed mean curvature in Euclidean spaces, I." J. Math. Soc. Japan 67 (3) 1077 - 1108, July, 2015. https://doi.org/10.2969/jmsj/06731077