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The minimal order of a first integral in the other direction is equal to $3M$ for an equation with the number $M$.\n  In the cases $M=1,\\ 2,\\ 3$ we show that those equations are integrable in quadratures. More precisely, we construct their general solutions in terms of the discrete integrals.\n  We also construct a modified series of Darboux"},"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":"1906.04503","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"nlin.SI","submitted_at":"2019-06-11T11:43:52Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"1dfdd44d3444ea3fe5a0a4fa6c0369e98a43cb2e5b9059ee0d2595fd42b66741","abstract_canon_sha256":"69d34bce0e3228b9926841ed43cad5e42fee8559c7cdd08373b30e7c2c44ab4d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:43:39.113335Z","signature_b64":"qcHIHXj+Xai8vBLYmXULIG8gvGd54hjLu0HSMyQQq9nAXTUx2g3k4Eipvt4s0nuWH4aj3yx9YYB9qSI6PQQODQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"11fc4dea5b41a677c218812c62304f4bacc85f42d5c1e246929dc6a70af41c06","last_reissued_at":"2026-05-17T23:43:39.112684Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:43:39.112684Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On a series of Darboux integrable discrete equations on the square lattice","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"nlin.SI","authors_text":"R.I. 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