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We show that $G$-equivariant counterexamples for infinite group $G$ exist if and only if $G=\\mathbb{C}^*$ and we classify them relating them to Belyi-Shabat polynomials. Taking universal covers we get rational simply connected $\\mathbb{C}^*$-surfaces of negative Kodaira dimension which admit non-proper $\\mathbb{C}^*$-equivariant \\'etale endomorphisms.\n  We prove also that for every integers $"},"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":"1701.01425","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2017-01-05T18:59:10Z","cross_cats_sorted":["math.AC","math.CV"],"title_canon_sha256":"cb38377461dd0d802d9f51bd6613de0b65b4f142a2f9302a46f2dce76e1be824","abstract_canon_sha256":"90aa0734ef438fc07ac5f23554643e21ebe2b1f5b9b0b25c7e0ee55c9807c369"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:47:36.572999Z","signature_b64":"DyRDru7c1q0Ma1M3UcTNYim0UZb+fVHhs6loKNIEYGENkGt5mwcrR7DilEcxmekKt6webDushXC1FOaQJ5U4Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c41cf4496d34b43fae60a99d2977bb34940f7ff6bcc43152315404fa40791d0f","last_reissued_at":"2026-05-17T23:47:36.572419Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:47:36.572419Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Jacobian Conjecture fails for pseudo-planes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC","math.CV"],"primary_cat":"math.AG","authors_text":"Adrien Dubouloz, Karol Palka","submitted_at":"2017-01-05T18:59:10Z","abstract_excerpt":"A smooth complex variety satisfies the Generalized Jacobian Conjecture if all its \\'etale endomorphisms are proper. 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