Darboux-Lamé equation and isomonodromic deformation (Q1773533)

The author, applying the results of his previous work [Osaka J. Math. 32, No.2, 409-430 (1995; Zbl 0849.34063)], considers the algebro-geometric Lamé operator \[ y''-(n(n+1)\varrho(z)-\lambda)y. \] He shows by using the so-called double algebro-geometric Darboux transformation, the possibility to construct a 1-parameter isomonodromic family of Darboux-Lamé differential equations of degenerated type. The last, in turn, by means of a classical change of variables, will be transformed into an isomonodromic family of equations over \(C(z)\). For the case \(n=2,\) all results are presented in an explicit form. For example, the second Darboux-Lamé equation of degenerated type has the form \[ y''=\frac{6\varrho(z)(\phi_{0}^2(\xi)-3\xi\phi_{0}(\xi)\varrho'(z)+(27/4)g_{3}\xi^{2})}{(\phi_{0}(\xi)+ (3/2)\xi\varrho'(z))^2}, \] where \(\xi \) is a parameter, \[ \phi_{0}(\xi)=(-(27/4) g_{3}\xi^{2}+c)^{1/2}, c \neq 0. \] For \(\xi=0\), the corresponding equation over \(C(z)\) has the following (hypergeometric) form \[ z(z-1)y''+\left(\frac{4}{3}z-\frac{2}{3}\right)y'-\frac{2}{3}y=0. \] For last equation the generators of the monodromy group are calculated.