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We say that $Y$ {\\it tracks} $X$ if $[Y,X]=fX$ for some continuous function $f\\colon M\\rightarrow\\mathbb F$. Let $K$ be a compact subset of the zero set ${\\mathsf Z}(X)$ such that ${\\mathsf\n  Z}(X)-K$ is closed, with nonzero Poincar\\'e-Hopf index (for example $K={\\mathsf Z}(X)$ when $M$ is compact and $\\chi(M)\\neq 0$) and let $\\mathcal G$ be a finite-dimensional Lie algebra of analytic vector fields on $M$. \\smallskip\n  {\\bf Theorem.} Let "},"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":"1606.08322","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2016-06-27T15:35:59Z","cross_cats_sorted":[],"title_canon_sha256":"b8c28a32bd9c0a43375260c703f2822dab6678528f891d94cd311243bcfef9c2","abstract_canon_sha256":"5a23014746e615f92910efade750c28466183000300bc4847d1c700be1f52a7a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:11:52.337483Z","signature_b64":"3OsodPBaS3doHFGaNpn2lAg1KvEeh85jXy4Kg96rh+S1zEBBWRlJUyf7kCSx+mrgHj6fld45SCOxGPlAeeV+DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"47b14cb923a2f0629fdcfb451a711687ba3a6ece20a50ca498b4774cee9fc576","last_reissued_at":"2026-05-18T01:11:52.337042Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:11:52.337042Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Zero sets of Lie algebras of analytic vector fields on real and complex 2-dimensional manifolds, II","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"F.-J. Turiel, Morris W. Hirsch","submitted_at":"2016-06-27T15:35:59Z","abstract_excerpt":"On a real ($\\mathbb F=\\mathbb R$) or complex ($\\mathbb F=\\mathbb C$) analytic connected 2-manifold $M$ with empty boundary consider two vector fields $X,Y$. We say that $Y$ {\\it tracks} $X$ if $[Y,X]=fX$ for some continuous function $f\\colon M\\rightarrow\\mathbb F$. Let $K$ be a compact subset of the zero set ${\\mathsf Z}(X)$ such that ${\\mathsf\n  Z}(X)-K$ is closed, with nonzero Poincar\\'e-Hopf index (for example $K={\\mathsf Z}(X)$ when $M$ is compact and $\\chi(M)\\neq 0$) and let $\\mathcal G$ be a finite-dimensional Lie algebra of analytic vector fields on $M$. \\smallskip\n  {\\bf Theorem.} Let "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.08322","kind":"arxiv","version":1},"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":"1606.08322","created_at":"2026-05-18T01:11:52.337113+00:00"},{"alias_kind":"arxiv_version","alias_value":"1606.08322v1","created_at":"2026-05-18T01:11:52.337113+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1606.08322","created_at":"2026-05-18T01:11:52.337113+00:00"},{"alias_kind":"pith_short_12","alias_value":"I6YUZOJDULYG","created_at":"2026-05-18T12:30:22.444734+00:00"},{"alias_kind":"pith_short_16","alias_value":"I6YUZOJDULYGFH64","created_at":"2026-05-18T12:30:22.444734+00:00"},{"alias_kind":"pith_short_8","alias_value":"I6YUZOJD","created_at":"2026-05-18T12:30:22.444734+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/I6YUZOJDULYGFH647NCRU4IWQ6","json":"https://pith.science/pith/I6YUZOJDULYGFH647NCRU4IWQ6.json","graph_json":"https://pith.science/api/pith-number/I6YUZOJDULYGFH647NCRU4IWQ6/graph.json","events_json":"https://pith.science/api/pith-number/I6YUZOJDULYGFH647NCRU4IWQ6/events.json","paper":"https://pith.science/paper/I6YUZOJD"},"agent_actions":{"view_html":"https://pith.science/pith/I6YUZOJDULYGFH647NCRU4IWQ6","download_json":"https://pith.science/pith/I6YUZOJDULYGFH647NCRU4IWQ6.json","view_paper":"https://pith.science/paper/I6YUZOJD","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1606.08322&json=true","fetch_graph":"https://pith.science/api/pith-number/I6YUZOJDULYGFH647NCRU4IWQ6/graph.json","fetch_events":"https://pith.science/api/pith-number/I6YUZOJDULYGFH647NCRU4IWQ6/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/I6YUZOJDULYGFH647NCRU4IWQ6/action/timestamp_anchor","attest_storage":"https://pith.science/pith/I6YUZOJDULYGFH647NCRU4IWQ6/action/storage_attestation","attest_author":"https://pith.science/pith/I6YUZOJDULYGFH647NCRU4IWQ6/action/author_attestation","sign_citation":"https://pith.science/pith/I6YUZOJDULYGFH647NCRU4IWQ6/action/citation_signature","submit_replication":"https://pith.science/pith/I6YUZOJDULYGFH647NCRU4IWQ6/action/replication_record"}},"created_at":"2026-05-18T01:11:52.337113+00:00","updated_at":"2026-05-18T01:11:52.337113+00:00"}