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Our result is:\n  {\\bf Theorem 1.} {\\em If $D$ has $P-$property, then $D$ is a ball.}\n  Four equivalent formulations of the Pompeiu problem are discussed. A domain $D$ has $P-$property if there exists an $f\\neq 0$, $f\\in L^1_{loc}(\\mathbb{R}^3)$ such that $\\int_{D}f(gx+y)dx=0$ for all $y\\in \\mathbb{R}^3$ and all $g\\in SO(2)$, where $ SO(2)$ is the rotation group. The result obtained concerning the related symmetry problem is:\n  {\\bf Theorem 2.} {\\em If $(\\nabla^2 +k^2)u=0$ in $D$, $u|_S=1$, $u_N|_S=0$, and $k>0$","authors_text":"A.G.Ramm","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-06-20T05:31:47Z","title":"Solution to the Pompeiu problem and the related symmetry problem"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.05976","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:96657563dc7b4f14e2c01c79d6e68b060919349ed3ca8ea45f911b51b8e80268","target":"record","created_at":"2026-05-18T01:09:06Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"f8b3af5035aceed0eb028128c9dae69e30b36c73788fbfe03bc9025469afea4d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-06-20T05:31:47Z","title_canon_sha256":"72f709f851694182fff9796018d1e9a5e9fe75c71b6d1233b40a041889daff02"},"schema_version":"1.0","source":{"id":"1606.05976","kind":"arxiv","version":2}},"canonical_sha256":"61fbb6e8b44a5b710ef1ae24b39afcd76c241cfe071f11b76d000f49be001c66","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"61fbb6e8b44a5b710ef1ae24b39afcd76c241cfe071f11b76d000f49be001c66","first_computed_at":"2026-05-18T01:09:06.306620Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:09:06.306620Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"6IB3haJ5384JSicMWWEGUIAItA1nD5Nl69lbkSUfSUUT+yBwe+0UylEKrmbhEin+qL8f6/M3hPO2DL5YlJ2WAg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:09:06.307088Z","signed_message":"canonical_sha256_bytes"},"source_id":"1606.05976","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:96657563dc7b4f14e2c01c79d6e68b060919349ed3ca8ea45f911b51b8e80268","sha256:d90ffab74d4a8d2eb03c4daa03097674bd4b7cbcb6507714f04f468138c5f25a"],"state_sha256":"bacff27cc69cdc0d1e5ae3a96ea8fbe7bbd6bf36552d48b791348507436d7697"}