  Let $\mathfrak{g}$ be a simple classical Lie algebra over $\mathbb{C}$ and
$G$ be the adjoint group. Consider a nilpotent element $e\in \mathfrak{g}$, and
the adjoint orbit $\mathbb{O}=Ge$. The formal slices to the codimension $2$
orbits in the closure $\overline{\mathbb{O}}\subset \mathfrak{g}$ are
well-known due to the work of Kraft and Procesi. In this paper, we prove a
similar result for the universal $G$-equivariant cover $\widetilde{\mathbb{O}}$
of $\mathbb{O}$. Namely, we describe the codimension $2$ singularities for its
affinization $Spec(\mathbb{C}[\widetilde{\mathbb{O}}])$.
