Ormerod, Christopher M. Symmetries in connection preserving deformations. (English) Zbl 1219.39005 SIGMA, Symmetry Integrability Geom. Methods Appl. 7, Paper 049, 13 p. (2011). Summary: We show that the root lattice of Bäcklund transformations of the \(q\)-analogue of the third and fourth Painlevé equations, which is of type \((A_{2}+A_{1})^{(1)}\), may be expressed as a quotient of the lattice of connection preserving deformations. Furthermore, we show that various directions in the lattice of connection preserving deformations present equivalent evolution equations under suitable transformations. These transformations correspond to the Dynkin diagram automorphisms. Cited in 1 Document MSC: 39A13 Difference equations, scaling (\(q\)-differences) 37K35 Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems 39A12 Discrete version of topics in analysis 34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies Keywords:Lax pairs; \(q\)-Schlesinger transformations; isomonodromy; Bäcklund transformations; Painlevé equations; connection preserving deformations; Dynkin diagram automorphisms PDFBibTeX XMLCite \textit{C. M. Ormerod}, SIGMA, Symmetry Integrability Geom. Methods Appl. 7, Paper 049, 13 p. (2011; Zbl 1219.39005) Full Text: DOI arXiv EuDML