# Verification report — AMR-154-0006 (Bolsinov–Konyaev–Matveev, Problem 4.3: conservation laws of the two-dimensional gl-regular Nijenhuis normal forms)

Verification date: 2026-10-18.

**Verdict.** The note answers both parts of the problem in the real analytic category; every statement it makes is
proved, and the scope is as follows.
- **Minimal version ("at least, find one regular conservation law for each of them").** Theorem 1.1 gives, for every
  normal form of Nijenhuis geometry III, Theorem 5.1 (all series, all admissible parameters) and for the three
  generic types, an explicit regular conservation law, with F_x(0,0) ≠ 0 computed in closed form. For the series
  O, P, S it is f_1(λ_1) + f_2(λ_2) (or 2 Re f(λ)) with f_i finite sums of negative fractional powers, or a
  logarithm, of the eigenvalue — the shape anticipated in the remark after the problem. Minimal exponents:
  −(2k−1)/(2k+1) (S^{2k+1}), −(k−1)/k (S^{2k}, and λ_1 in O), −(k−1)/(d−k) (λ_2 in O), −(s−1)/(2k) (P). The
  coefficients of the lower-order terms solve a triangular linear system with unit pivots; closed forms for k = 1.
- **Full version ("describe all conservation laws").** Theorem 1.2 describes all real-analytic conservation laws of
  every normal form: by direct integration for the generic types and L_nil, M, N; tr h(L) for L_nd; for O, P, S an
  explicit finite principal part subject to finitely many linear conditions (PC), at most one logarithm, and
  arbitrary analytic functions of explicit roots of the eigenvalues. Corollary 1.3: such a law is regular iff the
  coefficient of the most singular term (of the logarithm when N_0 = 0) is non-zero.
- **Known before.** The problem and the normal forms (Bolsinov, Konyaev, Matveev); the existence of a regular law
  near every point (Nijenhuis geometry IV, Theorem 1.5, an existence theorem in the real analytic category, not
  explicit; in dimension two it also follows directly from the Cauchy–Kovalevskaya theorem, since the coefficient
  of F_xx in E(F) is l_12 = 1 at a point where L = J_0); the laws at algebraically generic points, i.e. for the
  types G1–G3, among them L_nil and N, which are Jordan blocks at every point (Nijenhuis geometry IV, see the
  problem list, Section 4.1; NG IV, Theorem 1.4 treats a Jordan block whose eigenvalue is constant or has non-zero
  differential); at differentially non-degenerate points, in particular for L_nd, every law is tr F(L) (problem
  list, Section 4.1). These cases are included for completeness and credited; the new part is the explicit regular
  laws and the description of all analytic laws for the singular normal forms M, O, P, S.
- **Not treated.** The smooth category (where already existence is open), higher dimensions.
- **Literature caveat.** No printed answer was found in the sources searched. The survey A. V. Bolsinov,
  "Geodesically equivalent metrics and Nijenhuis geometry", Sb. Math. 216:5 (2025) 583–607, could not be read in
  full (host unreachable); its keywords include conservation laws, so an overlap cannot be excluded. No priority is
  claimed.

The note is unrefereed.

## Statement checked
- **Primary source.** A. V. Bolsinov, A. Yu. Konyaev, V. S. Matveev, "Research problems on relations between
  Nijenhuis geometry and integrable systems", arXiv:2410.04276v1 (5 October 2024), read in its LaTeX source.
  Published in 2024 MATRIX Annals, Part I, MATRIX Book Series, Springer, 2026, pp. 229–268,
  doi:10.1007/978-3-032-16202-1_8 (Crossref); the published chapter was not accessible to us.
  - The problem is the third `Problem` environment of Section 4 ("Nijenhuis operators: conservation laws and
    geodesically compatible metrics", subsection "Research problems"); the counter is per section, so it is
    Problem 4.3.
  - Section 4.1: all constructions are local and real analytic; f is a conservation law of L if L^*df is closed;
    a regular conservation law is one whose hierarchy f_1, …, f_n is functionally independent (in dimension two:
    df and L^*df independent at the point; Nijenhuis geometry IV, Section 1: df(p) cyclic for L^*(p)).
  - The remark after the problem expects the form f_1(λ_1) + f_2(λ_2) and asks for a non-zero differential; the
    note proves the stronger property (regularity).
- **Corpus record.** ulamai/UnsolvedMath 1.8.0, AMR-154-0006, "Nijenhuis Geometry and Integrable Systems — Problem 6"
  (the sixth problem extracted from the source list), status `open`; its statement agrees verbatim with the source.
- **Normal forms.** Nijenhuis geometry III, Rev. Mat. Iberoam. 40 (2024) 155–188, doi:10.4171/rmi/1416, Theorem 5.1
  (Section 5; Theorem 6.1 in Section 6 of arXiv:2007.09506v1), compared in both versions: L_nil, L_nd, M_{2k−1},
  M^ε_{2k}, N_{2k−1}, N^ε_{2k}; O^{d,ε}_{k,c}, P^{k,ε}_{s,c}, S^{2k,ε}_c, S^{2k+1}_c through the reconstruction formula
  (5.1), with the constants α as in the note; the generic types at the start of Section 5. The shift
  c_{k−1} → c_{k−1} + αx_0 along {y = 0} is in the paragraph between Remark 5.5 and Proposition 5.6 of the
  published version.

## Readings
| Reading | Answer | Where |
|---|---|---|
| Minimal version: one regular conservation law (df, L^*df independent) for each normal form, every parameter value | yes, explicit, with F_x(0,0) | Theorem 1.1, Table 1 |
| Weaker reading of the remark after the problem: d(f_1(λ_1) + f_2(λ_2)) ≠ 0 | implied (regularity is stronger) | Theorem 1.1 |
| Full version: all conservation laws (germs at the singular point) | complete description | Theorem 1.2 |
| "Each of these operators" includes the generic types of the classification | treated (known; included for completeness) | Table 1, Theorem 1.2(a) |
| Smooth category | not treated (the problem is posed in the real analytic category) | Section 1.4 |

## Results in the paper
- **Lemma 2.2 (eigenvalue functions).** For O, P, S (and L_nd) L^*dv = v dv + du, L^*du = u dv; a simple root λ
  satisfies L^*dλ = λ dλ; so h_1(λ_1) + h_2(λ_2) is a conservation law where the eigenvalues are distinct.
- **Lemma 2.3 (Hadamard).** With ξ = C(y) + αxy^{k−1}, a finite sum of terms y^{−j}H(ξ, y) extends analytically iff
  the Laurent series of its restriction to x = 0 has no principal part, and then F_x(0,0) is explicit.
- **Theorem 1.1 and Table 1.** The regular laws, with branch conventions stated under the table; the constants b_j
  by the recursion (PC) with unit pivots; closed forms for k = 1 (S^3: root of 8yΦ³ + 6Φ = c_0 + 3x; S^2: arsinh,
  arcosh, arccos; P^1: √(1+2εx) U_{s−2}(√(1+Z)); O_1: log(A/c_0)). Remark 3.1: for k ≥ 2 the b_j do not depend
  on c_{k−1}.
- **Theorem 1.2.** (a) G1: F_1(x) + F_2(y); G2: harmonic functions; G3, L_nil, N: A(y)E(x, f'(y)) + B(y); M: by
  characteristics; L_nd: tr h(L). (b) O, P, S: uniformising functions T_1, T_2 (Table 2), split types
  c log(T_1/T_2) + Σ(a_nT_1^{−n} + b_nT_2^{−n}) + g_1(T_1) + g_2(T_2), symmetric types
  Σ a_n(T_1^{−n} + T_2^{−n}) + g(T_1) + g(T_2), with (PC). Inclusion ⊇ by Lemmas 2.2–2.3; inclusion ⊆ by
  complexification, the uniformisation Ψ = (T_1, T_2) (injective, Section 4.5), the equation ∂²G/∂T_1∂T_2 = 0,
  a fibre lemma with a covering lemma and Cauchy estimates (|χ_i(t)| ≤ C|t|^{−M}), and monodromy / swap symmetry.
- **Section 4.6.** The fibre derivative D_1 = ∂_xT_1 − ∂_σT_1 ∂_xT_2/∂_σT_2 is analytic on a polydisc including σ = 0
  and equals κσ^M(1 + O(|x| + |σ|)) with M = N_0 + 1 and κ = 1, q, iq, c_0, 2εw(c_0), εw(c_0) for S^{2k+1},
  S^{2k} (real, complex), O, P (s odd, s even), computed in general.
- **Corollary 1.3.** F_x(0,0) = −N_0 κ a_{N_0} (N_0 ≥ 1), resp. κc (N_0 = 0).

## Computations (sanity checks; programs and outputs in reproducibility/)
No proof depends on a computation.
- **`check_note.py`** (written with the note; standard library, exact arithmetic over Q and Q(i), 21 s; 355
  checks, 0 failed): normal forms (112 parameter sets, k ≤ 4); Theorem 1.1 on the same 112 sets (box x^6 y^9) with
  the quoted b_j, Remark 3.1 and three negative controls; the k = 1 closed forms (38); the factorisation of D_1 with
  the constants of Table 2 (25); Theorem 1.2(b) and Corollary 1.3 on 5-jets (16; dimension 11 = 2N+1, spanned by
  the generators, not without the negative powers); Theorem 1.2(a).
- **Programs of the first version** (`original/`, SymPy): Theorem 1.1 as exact series to total degree 9 in 124
  instances (k ≤ 4); the k = 1 closed forms by exact simplification; 10 instances with numerically computed
  eigenvalues (50 digits); numerical fibre exponents (5 cases); jets of Theorem 1.2(b) in 23 instances (N = 7).
  Re-run on 2026-10-18 when the note was written: all seven outputs byte-identical to the recorded ones (about
  three minutes).
- **Verification run A** (`verification_run_A/`): 20 normal forms; 26 closed-form instances by algebraic reduction;
  29 series instances of Theorem 1.1 (k = 2, 3, 4); 34 instances of the eigenvalue formulas to 40 digits; the
  exponents and constants of Section 4.6 in 25 instances (10 S with x symbolic, 15 O/P on two slices of x); jets of
  Theorem 1.2(b) in 18 instances (N = 6). All OK.
- **Verification run B** (`verification_run_B/`): Theorem 1.1 in 140 instances with k ≤ 4 (boxes x^9 y^10 and
  x^12 y^14) and 20 with k = 5, 6, with negative controls; the closed forms and Lemma 2.2 symbolically; the
  factorisation of D_1 exactly in 32 instances; jets of Theorem 1.2(b) in 39 instances (N = 8); re-run of the seven
  programs of `original/` (byte-identical); the constants b_j of the first version recomputed in the 100 instances
  with rational square roots (100/100 equal). All OK. (The recorded `check_extra.out` ends with "ALL_OK False":
  for the M family with g = ±y³ SymPy did not simplify the residual F_x + gF_y − φ to zero; the same two cases are
  checked by `check_M_q3.py` at three points and as exact series, with residual 0.)
- **Verification run on the final text** (`independent_run_2/`; standard library and SymPy; written independently
  of all programs above): the eigenvalues as exact Puiseux series from the characteristic equation with the
  conventions of Table 1, the laws formed from them and the b_j by its own linear solve. Theorem 1.1 in 113 instances
  with k ≤ 4 (513 checks: normal forms, no poles, E(F) = 0 in boxes from x^2 y^13 to x^2 y^31, F_x(0,0), the b_1
  quoted after Table 1, Remark 3.1, negative controls); the k = 1 closed forms (38); Table 2, the factorisation of
  D_1, the rank of (PC), Corollary 1.3 and the jet count in 36 instances (24 with k ≤ 2 and J = 5, 12 with k = 3 and
  J = 4); Theorem 1.2(a) symbolically; the three k = 3 constants quoted in the package README. All OK.

## Independent verification runs
The results were first obtained with written proofs and the programs in `original/`. Two independent verification
runs, both AI-assisted, followed (2026-10-18); each re-derived the proofs and wrote its own programs, and neither
read the other's report. A third run examined the final text (next section).

| Item | Run A | Run B |
|---|---|---|
| Statement, definitions, list of normal forms against the sources | CONFIRMED (problem list LaTeX, both versions of NG III, NG IV) | CONFIRMED (problem list LaTeX, NG III arXiv and published) |
| Lemma 2.2, Lemma 2.3 | CONFIRMED (re-derived) | CONFIRMED (re-derived; Lemma 2.2 also symbolically) |
| Theorem 1.1 (all series, all parameters) | CONFIRMED (re-derived; 29 + 26 + 34 instances) | CONFIRMED (160 instances, k ≤ 6; closed forms symbolically) |
| Theorem 1.2(a) | CONFIRMED | CONFIRMED |
| Theorem 1.2(b), inclusion ⊇ | CONFIRMED | CONFIRMED |
| Theorem 1.2(b), inclusion ⊆ | CONFIRMED_WITH_FIXES: the fibre-derivative inputs to be written out for all types, with D_1 analytic at σ = 0 and the constants of the covering lemma | CONFIRMED_WITH_FIXES: the general factorisation D_1 = κσ^M(1 + O(|x|+|σ|)) to be written out |
| Literature, novelty | not checked beyond the claim (three abstracts) | nothing settling the question found; Bolsinov (2025) not read in full |

No run found a mathematical error.

**Corrections required by the runs**, all applied in the note:
1. (A1) The shift of c_{k−1}: cited from the paragraph between Remark 5.5 and Proposition 5.6 of the published
   NG III (Section 6 of arXiv v1), not from Remark 5.5: Remark 3.1.
2. (A2, B3) Section 4.6 written out in general for all types: D_1 analytic at σ = 0, factorisation with M and κ
   (Table 2), the bounds (II)–(IV), the covering lemma with explicit radii (|a|R/2), and the restriction
   |t| < ρ/6 so that the level set {T_2 = T_2^*} stays inside Ω (Section 4.4, Step 3).
3. (A3) Branch and root conventions stated under Table 1 (for P with εc_0 < 0: the eigenvalue with Im(ελ) > 0).
4. (A4) Akpan, Eur. J. Math. 8 (2022) (arXiv:2503.11852) studies the singular points with d tr L ≠ 0 but contains
   no conservation laws: so described in "Scope and priority" (the list of normal forms in that class was corrected
   after the run on the final text, see below).
5. (A5, B1) 124 instances, not 125.
6. (A6, optional) The regularity criterion: Corollary 1.3, with proof (Section 4.7).
7. (B2) Bolsinov (2025) stated as not accessible, with what was read (abstract, keywords, references) and the
   full-text-index phrases.
8. (B4) Journal references: NG IV, Nonlinearity 37 (2024) 105003; Applications V, J. Nonlinear Sci. 34 (2024) 33
   (Crossref).

**Written with the note, after the two runs.** Section 4.6 (from the runs' computations), Corollary 1.3, Remark 3.1
(the independence of b_j from c_{k−1}, checked in `check_note.py`), the merged Table 2 and `check_note.py`.

## Verification run on the final text (2026-10-18)
A third independent verification run, also AI-assisted, examined the note as written (13 pages before the
corrections below). It did not rely on the earlier reports.
- **Sources.** It read again, anonymously: the LaTeX source of the problem list (Section 4; Problem 4.3 is the third
  `Problem` environment of Section 4, counter per section; its text equals the corpus statement); NG III in the
  published version (Section 5, Theorem 5.1, formula (5.1), Remark 5.5, the shift paragraph, Proposition 5.6) and in
  arXiv v1 (the only version; Theorem 6.1); the LaTeX source of NG IV (Theorem 1.5 = `t6`, Theorem 1.4 = `jordan`,
  the definition of a regular conservation law, the remark on the smooth case); the abstract and LaTeX source of
  arXiv:2503.11852; the corpus records AMR-154-0001…0014 ("Problem 6" is the sixth record taken from the list). A
  further anonymous request to the host of the full text of Bolsinov (2025) timed out; the published NG IV and the
  published problem-list chapter were not accessible to us.
- **Proofs.** Every proof was re-derived line by line: the conventions of §1.2, every row of Table 1 with its branch
  conventions and F_x(0,0); Lemmas 2.1–2.3; the functions S_g, C_g; §3.2–3.5 and (PC); Remark 3.1; §4.1; §4.2–4.3;
  Lemma 4.1 and Steps 1–5 of §4.4 with all radii (ρ/2, |t| < ρ/6, |σ(x)| ≤ 6|t| < ρ, the bound
  (5/4)(e^{1/8} − 1) + 1/4 ≈ 0.416 < 1/2, the disc of radius |κ||σ_0|^M R/2, the Cauchy estimate); §4.5; §4.6 with
  all exponents and constants; §4.7. No mathematical error was found.
- **Computations.** Its own programs (`independent_run_2/`, described above) and a re-run of `check_note.py` (355
  checks, 0 failed, output identical) and of the seven programs of `original/` (outputs byte-identical).
- **Literature.** arXiv API (identifiers of all cited arXiv papers; Nijenhuis and conservation; Nijenhuis and
  singular; the authors Konyaev and Akpan, by date), Crossref (all six DOIs, verified), zbMATH Open (Nijenhuis and
  conservation laws; gl-regular and Nijenhuis), OpenAlex (only one work record could be retrieved), one web
  search. Nothing found answers the question; Bolsinov (2025) remains unread.
- **Corrections required, all made in the note:**
  1. Novelty and credit: L_nil and N (and G3) are algebraically generic (Jordan blocks at every point), where the
     problem list says the laws are described in NG IV; the statement of what is new (Sections 1.4 and 5) is now
     restricted to the explicit regular laws and the description of all analytic laws for M, O, P, S, and G1–G3,
     L_nil, N, L_nd are listed as known or elementary, included for completeness.
  2. Section 4.3: the terms of odd index in the symmetric types have, at x = 0, a pole of order *at most* (n−1)/2
     resp. (n−e)/2 (for S^{2k+1}_c with c_0 = 0 the order is smaller; for P with n < e there is no pole); the text
     said "exactly". The triangular structure of (PC), which uses only the even-index terms (order exactly n/2,
     leading coefficient 2), is unaffected.
  3. Akpan (2022): among the normal forms, the class d tr L ≠ 0 consists of L_nd, M_1, O and P with k = 1, and
     S^{2,ε}_c with c_0 ≠ 0 (not with c_0 = 0); the text said "the cases k = 1 of O, P and S^{2k}".
  4. This run recorded in the Verification paragraph and in this report; programs in `independent_run_2/`; the
     package README explains the "ALL_OK False" line of `verification_run_B/outputs/check_extra.out`.
  Optional changes made: Section 1.4 now says that in dimension two the existence of a regular law follows
  directly from the Cauchy–Kovalevskaya theorem (coefficient of F_xx in E(F) is l_12 = 1 at the point); §3.5
  writes ρ_−(ξ) = −1/ρ_+(ξ); the arXiv version of NG III is named as version 1.

## Relation to the literature, novelty and scope
- **Searches (18 October 2026).** arXiv API (Nijenhuis and conservation laws; the authors Bolsinov, Konyaev, Akpan,
  Manenti; "gl-regular"; Nijenhuis, singular, dimension two), Crossref, OpenAlex (work records, citing works,
  full-text probes), zbMATH Open, Semantic Scholar (rate-limited), six web searches (one of them by the run on the
  final text, which also repeated the arXiv, Crossref and zbMATH searches); the two works found to cite the
  problem list (arXiv:2411.02290, arXiv:2512.22064) read in abstract.
  - No answer to the question was found.
  - Akpan (2022) — singular points with d tr L ≠ 0, no conservation laws; Akpan–Bolsinov arXiv:2606.10751
    (Beltrami problem, 2D normal forms) and Akpan arXiv:2609.05103 (2D dictionary) — full texts have no
    conservation laws; Akpan–Manenti arXiv:2607.04913 — "conservation" only in a reference; BKM arXiv:2512.17609,
    arXiv:2604.03195 — conservation laws in general form only.
  - Bolsinov, Sb. Math. 216:5 (2025): abstract, keywords (including "conservation laws") and references read on a
    journal platform page; the full text could not be retrieved (mathnet.ru unreachable; a further attempt on
    18 October 2026 timed out). OpenAlex's full-text index reports the phrases "regular conservation law",
    "normal forms", "singular points" in it (not re-checked by the third run: OpenAlex refused its requests).
    zbMATH Open returns it for "Nijenhuis & conservation law*". Overlap not excluded.
- **What was read.** The problem list (Section 4, LaTeX source); NG III Section 5 (published) / Section 6 (arXiv v1);
  NG IV Section 1 (LaTeX source; theorem numbers of the arXiv version); the full texts and abstracts listed above.
  The proofs of the cited results were not checked; the note uses from NG III only the list of normal forms and the
  reconstruction formula, and does not use the proofs of NG IV.
- **Scope.** Real analytic category, dimension two, all normal forms. No priority is claimed. This negative search
  is not a proof of novelty.

## Public release and license
This paper, its source files, and this verification report are licensed under the Creative Commons Attribution
4.0 International License (CC BY 4.0): https://creativecommons.org/licenses/by/4.0/