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In this paper we extend the classification scheme in [S] for $b^m$-symplecticsurfaces and, more generally, $b^m$-Nambu structures to the equivariantsetting. When the compact group is the group of deck-transformations of anorientable covering, this yields the classification of these objects fornon-orientable surfaces. The paper also includes recipes to construct$b^m$-symplectic structures on surfaces. Feasibility of such constructionsdepends on orientability and on the colorability of an associated graph. Thedesingularization technique in [GMW] is revisited for surfaces and thecompatibility with this classification scheme is analyzed. We recast thestrategy used in [Mt] to classify stable Nambu structures of top degree onorientable manifolds to classify $b^m$-Nambu structures (not necessarilyoriented) using the language of $b^m$-cohomology. The paper ends up with anequivariant classification theorem of $b^m$-Nambu structures of top degree.

Equivariant classification of $b^m$-symplectic surfaces and Nambu structures