Baran and Marvan (Nonlinearity 23, 2010) classified Weingarten surfaces in Euclidean 3-space that are integrable in the sense of soliton theory, i.e. whose Gauss equation admits an sl(2)\mathfrak{sl}(2)-valued zero-curvature representation with a nonremovable spectral parameter. Under stated technical restrictions, the classified Weingarten relations are governed by a third-order ordinary differential equation solved by elliptic integrals; the generic one-parameter family yields the Gauss equation (39), a nonlinear PDE for ξ=ρ−σ\xi=\rho-\sigma with elliptic-integral coefficients, while the degenerate cases are the classical ones (linear Weingarten surfaces, surfaces of constant astigmatism, and other nineteenth-century classes). The paper explicitly left open the soliton-integrability structure of these equations: Bäcklund and Darboux transformations, recursion operators, bi-Hamiltonian structures, and higher generalized symmetries; explicit solutions beyond rotational seeds; the geometric meaning of the spectral parameter; the integrability of the focal surfaces and the induced Bianchi-type transformations; and extension of the classification beyond its technical assumptions. Subsequent literature (2011-2026) has developed these questions only for the degenerate constant astigmatism equation - for which bi-Hamiltonian structures, exact and multisoliton solutions, nonlocal conservation laws, and reciprocal transformations are now known - while the generic class, the spectral-parameter question, and the focal-surface questions remain open. The problem asks to establish the integrability structure of the classified equations, above all for the generic class.

Background

A smooth surface immersed in Euclidean 3-space is a Weingarten surface when its two principal curvatures k1,k2k_1,k_2 - equivalently the principal radii of curvature ρ=1/k1\rho=1/k_1, σ=1/k2\sigma=1/k_2 - satisfy a functional relation; locally one writes ρ=ρ(σ)\rho=\rho(\sigma). Minimal surfaces, surfaces of constant Gaussian or mean curvature, and their parallel surfaces are the classical examples. In curvature-line coordinates the fundamental forms of a surface are governed by the Gauss-Mainardi-Codazzi (GMC) equations, a nonlinear system for the metric coefficients; for a Weingarten surface the Gauss equation reduces to a single nonlinear partial differential equation, called the Gauss equation of the class.

In soliton theory, a partial differential equation is considered integrable when it admits a zero-curvature representation: a pair of matrices AA, BB depending on the field and finitely many of its derivatives such that Ay−Bx+[A,B]=0A_y-B_x+[A,B]=0 holds identically on solutions, and depending on a spectral parameter λ\lambda that cannot be removed by any gauge transformation (a nonremovable parameter). The spectral parameter is the gateway to the inverse scattering transform (the method that solves the initial-value problem by reformulating it as an associated linear scattering problem), to Bäcklund and Darboux transformations (solution-generating transformations that map solutions of the equation, or of a pair of such equations, to new solutions), and to algebro-geometric solutions. Marvan's method of the formal spectral parameter decides, cohomologically, when a parameter-free representation can be extended to one with a genuine spectral parameter; combined with the covering approach of Krasil'shchik and Marvan for GMC systems in special coordinates (a covering extends a differential system by auxiliary variables so that the original equations become differential consequences of the extended ones), it makes a systematic classification of integrable surface classes feasible, at the cost of heavy computer-algebra computations on jet spaces (spaces whose coordinates are the field variables and their derivatives up to a fixed order).

At the time, the prevailing expectation, based on the work of Wu and of Finkel, was that the only integrable Weingarten surfaces are the classical linear Weingarten ones, related by a linear relation between mean and Gaussian curvature. Baran and Marvan disproved this in Part I of their program by recovering a forgotten class of Beltrami, Bianchi, and Darboux: surfaces of constant astigmatism, defined by ρ−σ=const\rho-\sigma=\mathrm{const}, whose Gauss equation is the constant astigmatism equation zyy+(1/z)xx+2=0z_{yy}+(1/z)_{xx}+2=0. In Part II, the source of this problem, they completed the classification under technical restrictions: the representation takes values in sl(2,C)\mathfrak{sl}(2,\mathbb{C}), contains a fixed initial parameter-free representation, depends analytically on the parameter, and involves derivatives of no more than first order. The answer is a third-order ordinary differential equation for ρ(σ)\rho(\sigma) (eq. (10) of the paper) whose general solution, up to rescaling and the offsetting symmetry ρ→ρ+c0\rho\to\rho+c_0, σ→σ+c0\sigma\to\sigma+c_0 (passage to parallel surfaces), is given by elliptic integrals (eq. (34)). The generic family depends on one essential parameter cc and was believed by the authors to be new; when the elliptic integrals degenerate to elementary functions one recovers the classical classes: linear Weingarten surfaces, constant astigmatism, the relation ρ−σ=sin⁡(ρ+σ)\rho-\sigma=\sin(\rho+\sigma), Beltrami's involutes of constant-curvature surfaces, and tubular surfaces. For the generic family the Gauss equation (39) reads, for ξ=ρ−σ\xi=\rho-\sigma:

R′ ξyy+R′′ ξy2+S′ ξxx+S′′ ξx2−1ξ=0R'\,\xi_{yy}+R''\,\xi_y^{2}+S'\,\xi_{xx}+S''\,\xi_x^{2}-\frac{1}{\xi}=0

with

R′=1+c ξ2+Δξ2Δ,S′=c∓12 ξ2(1+c ξ2+Δ) Δ,Δ(ξ,c)=1+2 c ξ2+ξ4,R'=\frac{1+c\,\xi^{2}+\Delta}{\xi^{2}\Delta},\qquad S'=\frac{c\mp 1}{2}\,\frac{\xi^{2}}{(1+c\,\xi^{2}+\Delta)\,\Delta},\qquad \Delta(\xi,c)=\sqrt{1+2\,c\,\xi^{2}+\xi^{4}},

the sign matching the elliptic integral I±(ξ,c)=∫ξ(1±s2)/1+2 c s2+s4 dsI_\pm(\xi,c)=\int^{\xi}(1\pm s^2)/\sqrt{1+2\,c\,s^2+s^4}\,ds that fixes the class through ρ+σ=I±(ρ−σ,c)\rho+\sigma=I_\pm(\rho-\sigma,c) (eq. (38)).

The paper also characterized the normal line congruences of the classified surfaces. The focal surfaces (evolutes) r(1)=r+σ n\mathbf{r}^{(1)}=\mathbf{r}+\sigma\,\mathbf{n} and r(2)=r+ρ n\mathbf{r}^{(2)}=\mathbf{r}+\rho\,\mathbf{n} of an integrable Weingarten surface satisfy algebraic relations (47) between metric invariants G(i)G^{(i)} and κ(i)\kappa^{(i)} of the two focal surfaces, with constant product G(1)G(2)G^{(1)}G^{(2)}. For the constant astigmatism class, a classical theorem of Bianchi (1874), reproduced in the source paper, states that the focal surfaces are pseudospherical (of constant negative Gaussian curvature), and this is the mechanism by which Part I derived the integrability of that class. Whether an analogous integrability of focal surfaces holds for the generic class was left open.

The conclusion of the paper lists what could not be achieved. No Bäcklund or Darboux transformation, no recursion operator (an operator mapping generalized symmetries - transformations of the solution space depending on arbitrary functions - to new symmetries, thereby generating the infinite hierarchy), and no bi-Hamiltonian structure (a pair of compatible Hamiltonian operators whose recursion produces the commuting hierarchy of the equation) is known for the classified Gauss equations. No solutions of the generic Gauss equation beyond rotational seed surfaces were produced. The true geometric meaning of the spectral parameter in the constructed representations was unknown. Even the computation of third-order generalized symmetries proved too complex. The integrability of the focal surfaces of the generic class and the Bianchi-type transformations induced by relations (47) were unexplored. And the classification was not extended beyond its technical restrictions; its completeness is computer-aided, and the authors state they have no computer-independent proof of it.

Problem Statement

The problem is to work out the soliton-type integrability theory of the Weingarten Gauss equations classified by Baran and Marvan (2010), together with the geometric underpinning of that integrability - the package of results the source paper explicitly left open. All components attach to one and the same classified family of equations, and jointly they constitute 'knowing the integrability' of these equations in the standard soliton-theoretic sense, the sense in which the integrability of the sine-Gordon equation is considered worked out. Concretely, establish the following for the generic class and, where not already known, for the degenerate classes:

(1) Integrability attributes. Construct Bäcklund and/or Darboux transformations for the Gauss equations - at minimum for the generic Gauss equation (39) (the partial differential equation displayed in the Background above) - together with a recursion operator generating higher generalized symmetries and a bi-Hamiltonian (or at least Hamiltonian or Lagrangian) structure. All of these attributes are required, with either alternative sufficient where one is named; the recursion operator is what generates the third-order generalized symmetries of the generic Gauss equation, whose computation the authors could not complete.

(2) Explicit solutions. Produce nontrivial exact solutions of the generic Gauss equation (39) beyond the rotational seed solutions - for instance soliton-type or algebro-geometric solutions - together with the corresponding immersed surfaces reconstructed from the Gauss-Weingarten system.

(3) Geometric meaning of the spectral parameter. Elucidate what the nonremovable parameter λ\lambda in the constructed sl(2)\mathfrak{sl}(2) zero-curvature representations is, geometrically: for example, whether it is the parameter of a one-parameter transformation group of the extended linear problem, in the sense of the group-interpretation program of Cieśliński and co-workers, or whether varying λ\lambda performs a specified geometric transformation on the surfaces of the class - an associated family of isometric or conformally related surfaces, as the spectral parameter does for pseudospherical surfaces (Bäcklund transformations) and for isothermic surfaces (Calapso transformations, i.e. conformal rescalings coupled with a compensating isometry).

(4) Focal surfaces. Determine whether the focal surfaces of the generic class are integrable in their own right (admit zero-curvature representations with a nonremovable spectral parameter), which would explain the integrability of the class the way pseudospherical focal surfaces explain constant astigmatism; and construct and analyze the Bianchi-type transformation induced between surfaces whose focal-surface invariants satisfy relations (47).

(5) Robustness of the classification. Extend the classification beyond the technical assumptions under which it was obtained - representations beyond sl(2)\mathfrak{sl}(2), normal forms involving derivatives of higher than first order, and nonanalytic dependence on the spectral parameter - or show that these restrictions do not change the answer.

Scope notes: the paper formulates requests (1)-(3) 'in particular for the generic elliptic family and for special degenerations'; since the constant astigmatism degeneration has been substantially developed by later authors, the open core of (1)-(2) lies in the generic class and the remaining degenerations, while (3)-(5) are open for all classes. An answer relying only on the Sym-Tafel immersion formula (the construction of an immersed surface from a one-parameter family of solutions of the linear problem by differentiating with respect to the spectral parameter) still has to say what the resulting family of surfaces is geometrically. Completion of the problem is judged on the whole package for the generic class and, where not already known, for the degenerate classes, not on isolated components: contributions that leave the generic class's integrability attributes, its explicit solutions, the spectral parameter's geometric meaning, the focal-surface analysis, or the robustness of the classification undetermined do not complete it.

References

  1. Hynek Baran, Michal Marvan, Classification of integrable Weingarten surfaces possessing an sl(2)-valued zero curvature representation, Nonlinearity 23 (2010) 2577-2597. DOI: 10.1088/0951-7715/23/10/013; arXiv:1002.0992. https://arxiv.org/abs/1002.0992
  2. Hynek Baran, Michal Marvan, On integrability of Weingarten surfaces: a forgotten class, Journal of Physics A: Mathematical and Theoretical 42 (2009) 404007. DOI: 10.1088/1751-8113/42/40/404007; arXiv:1002.0989. https://arxiv.org/abs/1002.0989
  3. Hongyou Wu, Weingarten surfaces and nonlinear partial differential equations, Annals of Global Analysis and Geometry 11 (1993) 49-64. DOI: 10.1007/BF00773364. https://doi.org/10.1007/BF00773364
  4. F. Finkel, On the integrability of Weingarten surfaces, in: A. Coley et al. (eds.), Bäcklund and Darboux Transformations. The Geometry of Solitons, CRM Proceedings and Lecture Notes 29, American Mathematical Society, Providence, 2001, pp. 199-205. ISBN 978-0-8218-2803-8. https://www.ams.org/books/crmp/029/
  5. Michal Marvan, On the spectral parameter problem, Acta Applicandae Mathematicae 109 (2010) 239-255. DOI: 10.1007/s10440-009-9450-4; arXiv:0804.2031. https://arxiv.org/abs/0804.2031
  6. J. Krasil'shchik, M. Marvan, Coverings and integrability of the Gauss-Mainardi-Codazzi equations, Acta Applicandae Mathematicae 56 (1999) 217-230. DOI: 10.1023/A:1006121716159; arXiv:solv-int/9812010. https://arxiv.org/abs/solv-int/9812010
  7. Adam Hlaváč, Michal Marvan, Some results concerning the constant astigmatism equation, arXiv:1206.0321 (2012), preprint. https://arxiv.org/abs/1206.0321
  8. Adam Hlaváč, Michal Marvan, Another integrable case in two-dimensional plasticity, Journal of Physics A: Mathematical and Theoretical 46 (2013) 045203. DOI: 10.1088/1751-8113/46/4/045203. https://doi.org/10.1088/1751-8113/46/4/045203
  9. Maxim V. Pavlov, Sergej A. Zykov, Lagrangian and Hamiltonian structures for the constant astigmatism equation, Journal of Physics A: Mathematical and Theoretical 46 (2013) 395203. DOI: 10.1088/1751-8113/46/39/395203; arXiv:1212.6239. https://arxiv.org/abs/1212.6239
  10. Natale Manganaro, Maxim V. Pavlov, The constant astigmatism equation. New exact solution, Journal of Physics A: Mathematical and Theoretical 47 (2014) 075203. DOI: 10.1088/1751-8113/47/7/075203; arXiv:1311.1136. https://arxiv.org/abs/1311.1136
  11. Adam Hlaváč, Michal Marvan, A Reciprocal Transformation for the Constant Astigmatism Equation, SIGMA 10 (2014) 091. DOI: 10.3842/SIGMA.2014.091; arXiv:1111.2027. https://arxiv.org/abs/1111.2027
  12. Adam Hlaváč, Michal Marvan, On Lipschitz solutions of the constant astigmatism equation, Journal of Geometry and Physics 85 (2014) 88-98. DOI: 10.1016/j.geomphys.2014.05.020. https://doi.org/10.1016/j.geomphys.2014.05.020
  13. Adam Hlaváč, On multisoliton solutions of the constant astigmatism equation, Journal of Physics A: Mathematical and Theoretical 48 (2015) 365202. DOI: 10.1088/1751-8113/48/36/365202; arXiv:1503.06754. https://arxiv.org/abs/1503.06754
  14. Adam Hlaváč, Michal Marvan, Nonlocal conservation laws of the constant astigmatism equation, Journal of Geometry and Physics 113 (2017) 117-130. DOI: 10.1016/j.geomphys.2016.06.002; arXiv:1602.06861. https://arxiv.org/abs/1602.06861
  15. Adam Hlaváč, More exact solutions of the constant astigmatism equation, Journal of Geometry and Physics 123 (2018) 209-220. DOI: 10.1016/j.geomphys.2017.09.003; arXiv:1707.00775. https://arxiv.org/abs/1707.00775
  16. Shou-Fu Tian, Li Zou, Tian-Tian Zhang, Lie symmetry analysis, conservation laws and analytical solutions for the constant astigmatism equation, Chinese Journal of Physics 55 (2017) 1938-1952. DOI: 10.1016/j.cjph.2017.08.015. https://doi.org/10.1016/j.cjph.2017.08.015
  17. Rafael López, Álvaro Pámpano, Rotational surfaces of constant astigmatism in space forms, Journal of Mathematical Analysis and Applications 483 (2020) 123602. DOI: 10.1016/j.jmaa.2019.123602; arXiv:2005.07689. https://arxiv.org/abs/2005.07689
  18. Maxim V. Pavlov, Pierandrea Vergallo, Raffaele Vitolo, Classification of bi-Hamiltonian pairs extended by isometries, Proceedings of the Royal Society A 477 (2021) 20210185. DOI: 10.1098/rspa.2021.0185; arXiv:2011.04073. https://arxiv.org/abs/2011.04073
  19. Pierandrea Vergallo, Mats Vermeeren, Duality of Hamiltonian and Lagrangian formulations for integrable systems, arXiv:2604.19655 (2026), preprint. https://arxiv.org/abs/2604.19655
  20. Jan L. Cieśliński, Artur Kobus, Group interpretation of the spectral parameter. The case of isothermic surfaces, Journal of Geometry and Physics 113 (2017) 28-37. DOI: 10.1016/j.geomphys.2016.11.028; arXiv:1611.10148. https://arxiv.org/abs/1611.10148
  21. Jan L. Cieśliński, Dzianis Zhalukevich, Spectral Parameter as a Group Parameter, Symmetry 14 (2022) 2577. DOI: 10.3390/sym14122577. https://doi.org/10.3390/sym14122577
  22. Jan L. Cieśliński, Artur Kobus, Lax Triples for Integrable Surfaces in Three-Dimensional Space, Advances in Mathematical Physics 2016 (2016) 8386420. DOI: 10.1155/2016/8386420. https://doi.org/10.1155/2016/8386420
  23. Michal Marvan, On Integrable Nets in General and Concordant Chebyshev Nets in Particular, SIGMA 21 (2025) 029. DOI: 10.3842/SIGMA.2025.029; arXiv:2403.12626. https://arxiv.org/abs/2403.12626
  24. Nikos Georgiou, Brendan Guilfoyle, Morgan Robson, Isotropic submanifolds of TSnT\mathbb{S}^n and their focal sets, arXiv:2601.17451 (2026), preprint. https://arxiv.org/abs/2601.17451