  We establish an injective correspondence $M\longrightarrow\mathcal E(M)$
between real-analytic nonminimal hypersurfaces $M\subset\mathbb{C}^{2}$,
spherical at a generic point, and a class of second order complex ODEs with a
meromorphic singularity. We apply this result to the proof of the bound
$\mbox{dim}\,\mathfrak{hol}(M,p)\leq 5$ for the infinitesimal automorphism
algebra of an \it arbitrary \rm germ $(M,p)\not\sim(S^3,p')$ of a real-analytic
Levi nonflat hypersurface $M\subset\mathbb{C}^2$ (the Dimension Conjecture).
This bound gives the first proof of the dimension gap
$\mbox{dim}\,\mathfrak{hol}(M,p)=\{8,5,4,3,2,1,0\}$ for the dimension of the
automorphism algebra of a real-analytic Levi nonflat hypersurface. As another
application we obtain a new regularity condition for CR-mappings of nonminimal
hypersurfaces, that we call \it Fuchsian type, \rm and prove its optimality for
extension of CR-mappings to nonminimal points. \\ We also obtain an existence
theorem for solutions of a class of singular complex ODEs (Theorem 3.5).
