On spherical type singularities in integrable systems

This thesis is devoted to the study of a relatively new class of integrable systems, characterized by singular fibers in the Lagrangian fibration that one might describe as being of 'spherical type' - that is they are smooth submanifolds diffeomorphic to product of spheres of any dimension, although possibly quotient by the action of some finite group. In these systems we have globally defined and continuous but not necessarily smooth action variables. These fibers of special interest appear as fibers over points where the actions are not smooth. We study certain examples of these systems. For geodesic flow on the sphere Sⁿ (with integrals associated to Vilenkin's polyspherical coordinates) we give a complete description of the moment cone as well as a topological description of all singular fibers. We then study the system of bending flows on polygon spaces, proving various results centred on describing the topology of fibers over certain simplicial vertices of the moment polytope. Finally, we verify that geodesic flow on S² can be used as a local model for a neighbourhood of the S² singular fibers appearing in systems of bending flows on spaces of pentagons.