This volume, whose contributors include leading researchers in their field, covers a wide range of topics surrounding Integrable Systems, from theoretical developments to applications. Comprising a unique collection of research articles and surveys, the book aims to serve as a bridge between the various areas of Mathematics related to Integrable Systems and Mathematical Physics.

Victor Buchstaber, Sotiris Konstantinou-Rizos and Alexander Mikhailov, Introduction.- P. G. Grinevich and P. M. Santini, Numerical instability of the Akhmediev breather and a finite-gap model of it.- V.Yu. Novokshenov, Movable poles of Painlevé I transcendents and singularities of monodromy data manifolds.- Vladimir Sokolov, Elliptic Calogero-Moser Hamiltonians and compatible Poisson brackets.- A.V. Tsiganov, Bäcklund transformations and new integrable systems on the plane.- V. Rubtsov, Quadro-cubic Cremona transformations and Feigin-Odesskii-Sklyanin algebras with 5 generators.- Dimitri Gurevich and Pavel Saponov, From Reflection Equation Algebra to Braided Yangians.- G. Sharygin and A. Konyaev, Survey of the Deformation Quantization of Commutative Families.- Dmitry Millionshchikov, Graded Thread Modules over the Positive Part of the Witt (Virasoro) Algebra.- Andrei Shafarevich, Localized solutions of the Schrödinger equation on hybrid spaces. Relation to the behavior of geodesics andto analytic number theory.- A. V. Fursikov, Normal equation generated from Helmholtz system: nonlocal stabilization by starting control and properties of stabilized solutions.