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Under some orientability hypothesis, we give a necessary condition for the existence of a plane field $\\mathcal{D}^2$ such that $\\mathcal{W}\\subset\\mathcal{D}$ and $[\\mathcal{D},\\mathcal{D}]=\\mathcal{E}$. Moreover we study the case where a section of $\\mathcal{W}$ is non-singular Morse-Smale and we get a sufficient condition for the global existence of $\\mathcal{D}$. As a corollary we get con"},"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":"1905.11839","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2019-05-28T14:23:21Z","cross_cats_sorted":[],"title_canon_sha256":"21bdfa1242b06c729ee3861dfd27926b23ff07e472e8872b9514a54b52a4f41e","abstract_canon_sha256":"5a37a04edc2d99a6335885b36ec6e017956b6541c54686ec67287a555e2222b3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:44:50.122838Z","signature_b64":"Ad038GD+pgHZfNeoJ+h3gMnuaxD0p03Rl03CvTjBNUCgVpzzp93kCyd32/+m24bsmwh/Qjp864iI/7mNuyk8BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"62a7a4a25b191f392009e3caae76ab6f01dc99deca157710bae8cdb0b53f05e9","last_reissued_at":"2026-05-17T23:44:50.122273Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:44:50.122273Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the dynamics of some vector fields tangent to non-integrable plane fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Nicola Pia","submitted_at":"2019-05-28T14:23:21Z","abstract_excerpt":"Let $\\mathcal{E}^3\\subset TM^n$ be a smooth $3$-distribution on a smooth manifold of dimension $n$ and $\\mathcal{W}\\subset\\mathcal{E}$ a line field such that $[\\mathcal{W},\\mathcal{E}]\\subset\\mathcal{E}$. 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