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Let $H$ be some isotropy subgroup for this action representing the principle orbit type and $X^r_\\mathfrak{h}$ be the submanifold of $X$ consisting of the points in $X$ with the stabilizer algebra equal to the Lie algebra $\\mathfrak{h}$ of $H$ and with the stabilizer group conjugated to $H$ in $G$. We prove that the pair of symplectic structures $\\omega_1|_{X^r_\\mathfrak{h}}$ and $\\omega_2|_{X^r_\\mathf"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1605.03382","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2016-05-11T11:30:35Z","cross_cats_sorted":[],"title_canon_sha256":"3697d75791a8a232269aaa5236f65d77cbdcf2973b1f8e76423dfa2bc2a1897e","abstract_canon_sha256":"42f646cea188d31e965095bfa5eb517735d59cfc7da957c6a8a3accfc26b54e2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:11:01.797233Z","signature_b64":"+5aNGspjVzlJ22ngCkmkrPGBbawuNGu0j3olNNd4DyQ+kbZzWtAX7Cnt7BT6cEbnkzt3WHn3A4nl7z40AKjqBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e94966475bde90f36e26c0b6b34fac269ba3cd67216fab799e9470d591ffb8ad","last_reissued_at":"2026-05-18T01:11:01.796822Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:11:01.796822Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Dirac brackets and reduction of invariant bi-Poisson structures","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Andriy Panasyuk, Ihor V. Mykytyuk","submitted_at":"2016-05-11T11:30:35Z","abstract_excerpt":"Let $X$ be a manifold with a bi-Poisson structure $\\{\\eta^t\\}$ generated by a pair of $G$-invariant symplectic structures $\\omega_1$ and $\\omega_2$, where the Lie group $G$ acts properly on $X$. Let $H$ be some isotropy subgroup for this action representing the principle orbit type and $X^r_\\mathfrak{h}$ be the submanifold of $X$ consisting of the points in $X$ with the stabilizer algebra equal to the Lie algebra $\\mathfrak{h}$ of $H$ and with the stabilizer group conjugated to $H$ in $G$. We prove that the pair of symplectic structures $\\omega_1|_{X^r_\\mathfrak{h}}$ and $\\omega_2|_{X^r_\\mathf"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.03382","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1605.03382","created_at":"2026-05-18T01:11:01.796886+00:00"},{"alias_kind":"arxiv_version","alias_value":"1605.03382v2","created_at":"2026-05-18T01:11:01.796886+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1605.03382","created_at":"2026-05-18T01:11:01.796886+00:00"},{"alias_kind":"pith_short_12","alias_value":"5FEWMR2332IP","created_at":"2026-05-18T12:30:01.593930+00:00"},{"alias_kind":"pith_short_16","alias_value":"5FEWMR2332IPG3RG","created_at":"2026-05-18T12:30:01.593930+00:00"},{"alias_kind":"pith_short_8","alias_value":"5FEWMR23","created_at":"2026-05-18T12:30:01.593930+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/5FEWMR2332IPG3RGYC3LGT5ME2","json":"https://pith.science/pith/5FEWMR2332IPG3RGYC3LGT5ME2.json","graph_json":"https://pith.science/api/pith-number/5FEWMR2332IPG3RGYC3LGT5ME2/graph.json","events_json":"https://pith.science/api/pith-number/5FEWMR2332IPG3RGYC3LGT5ME2/events.json","paper":"https://pith.science/paper/5FEWMR23"},"agent_actions":{"view_html":"https://pith.science/pith/5FEWMR2332IPG3RGYC3LGT5ME2","download_json":"https://pith.science/pith/5FEWMR2332IPG3RGYC3LGT5ME2.json","view_paper":"https://pith.science/paper/5FEWMR23","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1605.03382&json=true","fetch_graph":"https://pith.science/api/pith-number/5FEWMR2332IPG3RGYC3LGT5ME2/graph.json","fetch_events":"https://pith.science/api/pith-number/5FEWMR2332IPG3RGYC3LGT5ME2/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5FEWMR2332IPG3RGYC3LGT5ME2/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5FEWMR2332IPG3RGYC3LGT5ME2/action/storage_attestation","attest_author":"https://pith.science/pith/5FEWMR2332IPG3RGYC3LGT5ME2/action/author_attestation","sign_citation":"https://pith.science/pith/5FEWMR2332IPG3RGYC3LGT5ME2/action/citation_signature","submit_replication":"https://pith.science/pith/5FEWMR2332IPG3RGYC3LGT5ME2/action/replication_record"}},"created_at":"2026-05-18T01:11:01.796886+00:00","updated_at":"2026-05-18T01:11:01.796886+00:00"}