  We consider an example of a system with two degrees of freedom admitting
separation of variables but having a subset of codimension 1 on which the
2-form defining the symplectic structure degenerates. We show how to use
separation of variables to calculate the exact topological invariant of
non-degenerate singularities and singularities appearing due to the symplectic
structure degeneration. New types of non-orientable 3-atoms are found.
