  We study generic constrained differential equations (CDEs) with three
parameters, thereby extending Takens's classification of singularities of such
equations. In this approach, the singularities analyzed are the Swallowtail,
the Hyperbolic, and the Elliptic Umbilics. We provide polynomial local normal
forms of CDEs under topological equivalence. Generic CDEs are important in the
study of slow-fast (SF) systems. Many properties and the characteristic
behavior of the solutions of SF systems can be inferred from the corresponding
CDE. Therefore, the results of this paper show a first approximation of the
flow of generic SF systems with three slow variables.
