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We introduce GBDT version of Darboux transformation for symplectic andHamiltonian systems as well as for Shin-Zettl systems and Sturm-Liouvilleequations. These are the first results on Darboux transformation forgeneral-type Hamiltonian and for Shin-Zettl systems. The obtained results areapplied to the corresponding transformations of the Weyl-Titchmarsh functionsand to the construction of explicit solutions of dynamical symplectic systems,of two-way diffusion equations and of indefinite Sturm-Liouville equations. Theenergy of the explicit solutions of dynamical systems is expressed (in a quitesimple form) in terms of the parameter matrices of GBDT.

Hamiltonian systems and Sturm-Liouville equations: Darboux transformation and applications