{'fraction_non_alphanumeric': 0.06376811594202898, 'fraction_numerical': 0.01642512077294686, 'mean_word_length': 4.954022988505747, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Reductions of Invariant bi-Poisson Structures and Locally Free Actions

Let (X,G,ω1,ω2,{ηt}) be a manifold with a bi-Poisson structure {ηt} generated by a pair of G-invariant symplectic structures ω1 and ω2, where a Lie group G acts properly on X. We prove that there exists two canonically defined manifolds (RLi,Gi,ω1i,ω2i,{ηit}), i=1,2 such that (1) RLi is a submanifold of an open dense subset X(H)⊂X; (2) symplectic structures ω1i and ω2i, generating a bi-Poisson structure {ηit}, are Gi- invariant and coincide with restrictions ω1|RLi and ω2|RLi; (3) the canonically defined group Gi acts properly and locally freely on RLi; (4) orbit spaces X(H)/G and RLi/Gi are canonically diffeomorphic smooth manifolds; (5) spaces of G-invariant functions on X(H) and Gi-invariant functions on RLi are isomorphic as Poisson algebras with the bi-Poisson structures {ηt} and {ηit} respectively. The second Poisson algebra of functions can be treated as the reduction of the first one with respect to a locally free action of a symmetry group.