Topology of algebraically separable integrable systems

We classify the simplest 3-dimensional singularities of regular algebraically separable integrable systems. Such systems form an important class of Liouville integrable Hamiltonian systems with two degrees of freedom and occur in many problems of mechanics and geometry. The techniques elaborated in the paper is based on the analysis of a certain $\mathbb Z_2$-matrix uniquely determined by the expressions of the initial phase variables via the separating variables.