E-structures and almost regular Poisson manifolds

In recent years, $b$-symplectic manifolds have become important structures in the study of symplectic geometry, serving as Poisson manifolds that retain symplectic properties away from a hypersurface. Inspired by this rich landscape, $E$-structures were introduced by Nest and Tsygan in \cite{NT2} as a comprehensive framework for exploring generalizations of $b$-structures. This paper initiates a deeper investigation into their Poisson facets, building on foundational work by \cite{MS21}. We also examine the closely related concept of almost regular Poisson manifolds, as studied in \cite{AZ17}, which reveals a natural Poisson groupoid associated with these structures. In this article, we investigate the intricate relationship between $E$-structures and almost regular Poisson structures. Our comparative analysis not only scrutinizes their Poisson properties but also offers explicit formulae for the Poisson structure on the Poisson groupoid associated to the $E$-structures as both Poisson manifolds and singular foliations. In doing so, we reveal an interesting link between the existence of commutative frames and Darboux-Carath\'eodory-type expressions for the relevant structures.