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In this paper we deal with the problem of construction of second order regular differential equations with polynomial coefficients on Riemann sphere with given monodromy, such problem arises from inverse problem of electrostatics and is related to controlability of quantum systems. We consider the particular case of this problem. [ABSTRACT FROM AUTHOR]

A new scheme for investigating the analytical properties of the solution of the n-th stationary KdV equation is observed with the aid of its algebraic first integrals in involution. For instance, the double periodicity of the solution of the n-th stationary KdV equation is clarified by calculating the monodromy of the differential equation with the regular singular points which are derived from the eigenvalue problem associated with the linearizing operator of the n-th stationary KdV equation. The role of the spectral parameter for the resulting Fuchsian equation is also observed. [ABSTRACT FROM AUTHOR]

We review Darboux-Crum transformation of Heun’s differential equation. By rewriting an integral transformation of Heun’s differential equation into a form of elliptic functions, we see that the integral representation is a generalization of Darboux-Crum transformation. We also consider conservation of monodromy with respect to the transformations. [ABSTRACT FROM AUTHOR]

From a factorization formula for the monodromy element of a periodic Lie system on a Lie group, we show that the concepts geometric and dynamical phases are naturally defined for such class of systems. An interpretation of the phases is given in terms of the Poisson geometry and the Hamiltonian dynamics induced by the coadjoint action on the Lie coalgebra of the group. Applying these results we give a general formulae for the reconstruction dynamics of a system with symmetry which generalize previous results given by Marsden et al. in [1]. [ABSTRACT FROM AUTHOR]

Here we present some necessary and sufficient conditions for the validity of the Saxon-Hutner conjecture concerning the preservation of the energy gaps into an infinite one-dimensional lattice. [ABSTRACT FROM AUTHOR]

Wilson's interpretation of the Miura map ν->u as being part of a chain of field extensions C⊂C<ν>⊂C<[lowercase_phi_synonym],ψ> is revisited. By (i) incorporating monodromy in the space of solutions to the linear problem ψ″+uψ = 0 and (ii) extending the notion of Poisson symmetry to those of Poisson Lie group type, the result of Wilson is shown to be an example of Poisson reduction. The result is extended to the lattice analogue by direct means. [ABSTRACT FROM AUTHOR]