Using a Lax pair based on twisted affine sl(2, R) Kac-Moody and Virasoro algebras, we deduce a r-matrix formulation of two dimensional reduced vacuum Einstein’s equations. Whereas the fundamental Poisson brackets are non-ultralocal, they lead to pure c-number modified Yang-Baxter equations. We also describe how to obtain classical observables by imposing reasonable boundaries conditions. Member...

We show how the elliptic Calogero-Moser integrable systems arise from a symplectic quotient construction, generalising the construction for AN−1 by Gorsky and Nekrasov to other algebras. This clarifies the role of (twisted) affine Kac-Moody algebras in elliptic Calogero-Moser systems and allows for a natural geometric construction of Lax operators for these systems. We elaborate on the connecti...

We combine two research directions of the past decade, namely the development of a lax-algebraic framework for categories of interest to topologists and analysts, and the exploration of key topological concepts, like separation and compactness, in an abstract category which comes equipped with an axiomatic notion “closed” or “proper” map. Hence, we present various candidates for such notions in...

For each one of the Lie algebras gln and g̃ln, we constructed a family of integrable generalizations of the Toda chains characterized by two integers m+ and m−. The Lax matrices and the equations of motion are given explicitly, and the integrals of motion can be calculated in terms of the trace of powers of the Lax matrix L. For the case of m+ = m−, we find a symmetric reduction for each general...

Calogero-Moser models can be generalised for all of the finite reflection groups. These include models based on non-crystallographic root systems, that is the root systems of the finite reflection groups, H3, H4, and the dihedral group I2(m), besides the well-known ones based on crystallographic root systems, namely those associated with Lie algebras. Universal Lax pair operators for all of the...

We consider the 4D Martínez Alonso–Shabat equation E uty=uzuxy−uyuxz (also referred to as universal hierarchy equation) and using its known Lax pair construct two infinite-dimensional differential coverings over E. In these coverings, we give a complete description of Lie algebras nonlocal symmetries. particular, our results generalize ones obtained in Morozov Sergyeyev (2014) contain construct...

The Drinfeld-Sokolov reduction method has been used to associate with gln extensions of the matrix r-KdV system. Reductions of these systems to the fixed point sets of involutive Poisson maps, implementing reduction of gln to classical Lie algebras of type B, C, D, are here presented. Modifications corresponding, in the first place to factorisation of the Lax operator, and then to Wakimoto real...

The algebraic structure and the spectral properties of a special class of multicomponent NLS equations, related to the symmetric spaces of BD.I-type are analyzed. The focus of the study is on the spectral theory of the associated Lax operator to these nonlinear evolutionary equations for different fundamental representations of the underlying simple Lie algebra g. Special attention is paid to t...

We consider the 3D equation $u_{yy} = u_{tx} + u_yu_{xx} - u_xu_{xy}$ and its 2D reductions: (1) (u_y+y)u_{xx}-u_xu_{xy}-2$ (which is equivalent to Gibbons-Tsarev equation) (2) (u_y+2x)u_{xx} (y-u_x)u_{xy} -u_x$. Using reduction of known Lax pair for equation, we describe nonlocal symmetries of~(1) and~(2) show that Lie algebras these are isomorphic Witt algebra.