Sergyeyev, A. Infinite hierarchies of nonlocal symmetries of the Chen-Kontsevich-Schwarz type for the oriented associativity equations. (English) Zbl 1186.37036 J. Phys. A, Math. Theor. 42, No. 40, Article ID 404017, 15 p. (2009). The author of this interesting paper constructs infinite hierarchies of nonlocal higher symmetries for the oriented associativity equations using for this purpose the solutions of auxiliary spectral problems. The structure constants \(c_{\alpha\beta }^{\delta }(x^1,...,x^n)\) of a commutative algebra satisfy the relations \[ c_{\alpha\rho }^{\nu }c_{\beta\gamma }^{\rho }=c_{\rho \gamma }^{\nu }c_{\alpha\beta }^{\rho }, \;\;{\partial c_{\beta\gamma }^{\alpha }\over \partial x^{\rho }}={\partial c_{\rho\gamma }^{\alpha }\over \partial x^{\beta }}. \] The symmetries mentioned above generalize those found by Chen, Kontsevich and Schwarz for the Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations. An interesting result that the author obtains is a Darboux-type transformation and a conditional Bäcklund transformation for the oriented associativity equation. As a conclusion one could find some open problems arising in the theory. Reviewer: Dimitar A. Kolev (Sofia) Cited in 8 Documents MSC: 37C80 Symmetries, equivariant dynamical systems (MSC2010) 37K35 Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems 37K05 Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010) 37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems Keywords:hierarchies; symmetries; Chen-Kontsevich-Schwarz; Darboux-type transformation; Frobenius manifolds; Hessian reduction PDFBibTeX XMLCite \textit{A. Sergyeyev}, J. Phys. A, Math. Theor. 42, No. 40, Article ID 404017, 15 p. (2009; Zbl 1186.37036) Full Text: DOI arXiv