Categories of Symmetry Groups of the Space of Solutions of the Special Doubly Confluent Heun Equation

The representations of the groups $G_{\rm I}$, $G_{\rm II}$, $G_{\rm III}$, $G_{\rm IV}$ that characterize symmetries of the solution space of a special doubly confluent Heun equation are described. Categories of groups whose commutator subgroup is isomorphic to the group of integers are introduced, and an algorithm for categorical characterization of such groups is described. An implementation of the algorithm for the groups $G_{\rm I}$, …, $G_{\rm IV}$ is given.