Invariants in almost Hamiltonian systems with non-orientable phase space

The topological analysis of irreducible systems with three degrees of freedom is based on the investigation of the critical sets of momentum maps. Such set is a union of phase spaces of integrable systems (critical subsystems) with less number of degrees of freedom usually having subsets of codimension 1 on which the induced symplectic structure degenerates. We consider a real example from the rigid body dynamics of a critical subsystem on a non-orientable 4-manifold. We show how to calculate the gluing matrices using the separation of variables and find, in the neighborhood of the set of symplectic form degeneration, new atoms which are obtained as quotient manifolds of standard 3-atoms with respect to a $\mathbb{Z}_2$-symmetry.