In this paper we analyze in detail a collection of motivating examples to consider bm-symplectic forms and folded-type symplectic structures. In particular, we provide models in Celestial Mechanics for every bm-symplectic structure. At the end of the paper, we introduce the odd-dimensional analogue to b-symplectic manifolds: b-contact manifolds.
Miranda, E., Delshams, A., Dempsey, R., Oms, C., Planas, A. An invitation to singular symplectic geometry. "International journal of geometric methods in modern physics", vol.16. supp.01, 2019. Art. 1940008