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In this talk, we mainly consider some issues related to the study of Volterra lattice (also known as Kac-van Moerbeke system, Langmuir chain or discrete Korteweg-de Vries equation) in the semi-infinite case. In particular, we consider its integration via Lax pair formalism by means of the inverse spectral problem for (Jacobi-like) second order difference operators. The key role in this inverse problem integration method is played by the moments of the Weyl function of the corresponding difference operator, which appears in the Lax representation for this lattice, and their evolution in time. We discuss the extension of this method to another classes of nonlinear dynamical systems (e.g. Bogoyavlensky lattices) and their other applications to the theory of nonlinear integrable equations. This talk is partly based on the following papers:

[1] Osipov A.S. (2020) Inverse spectral problems for second-order difference operators and their application to the study of Volterra type systems, Rus. J. Nonlin. Dyn., 16:3, 397--419.

[2] Osipov A. S. (2021) Inverse spectral problem for Jacobi operators and Miura transformation, Concr. Oper., :1, 77--89.

[3] Osipov A. S. (2022) Inverse spectral problem for band operators and their sparsity criterion in terms of inverse problems data, Russian Journal of Mathematical Physics, 29:2, 225--237