  We consider the nonlinear Sturm-Liouville differential operator $F(u) = -u''
+ f(u)$ for $u \in H^2_D([0, \pi])$, a Sobolev space of functions satisfying
Dirichlet boundary conditions. For a generic nonlinearity $f: \RR \to \RR$ we
show that there is a diffeomorphism in the domain of $F$ converting the
critical set $C$ of $F$ into a union of isolated parallel hyperplanes. For the
proof, we show that the homotopy groups of connected components of $C$ are
trivial and prove results which permit to replace homotopy equivalences of
systems of infinite dimensional Hilbert manifolds by diffeomorphisms.
