We consider a generalization of the Camassa Holm (CH) equation with two dependent variables, called CH2, introduced in [16]. We briefly provide an alternative derivation of it based on the theory of Hamiltonian structures on (the dual of) a Lie Algebra. The Lie Algebra here involved is the same algebra underlying the NLS hierarchy. We study the structural properties of the CH2 hierarchy within the bihamiltonian theory of integrable PDEs, and provide its Lax representation. Then we explicitly discuss how to construct classes of solutions, both of peakon and of algebro-geometrical type. We finally sketch the construction of a class of singular solutions, defined by setting to zero one of the two dependent variables.formulation by means of a τ -function. This lacking is reflected in the properties of notable classes of solutions. Indeed, bounded traveling waves for the CH equation (termed peakons) develop a discontinuity in the first derivatives, and the evolution properties of finite gap solutions, as discussed in [4,17,5], are somewhat peculiar, even if they can be expressed in terms of hyperelliptic curves as in the KdV case. The Whitham modulation theory associated with genus one solutions of CH, discussed in [1], also presents some non standard features.The integrable equations we are going to discuss in the present paper are conservation laws for two dependent variables of the form:( 1.1) with ρ a parameter. These equations was derived (for ρ = 1) in [16] within the framework of the general deformation theory of hydrodynamic hierarchies of bihamiltonian evolutionary PDEs [9]; more recently, a related equation has been considered in [7], within the framework of reciprocal transformations, and the properties of solitary waves and 2-particle like solutions described. Actually, a similar equation was introduced by Olver and Rosenau in [18], as a deformation of the Boussinesq system; traveling wave solutions of the resulting equation can be found in [15]. In particular, in the paper [18], various non-standard integrable equations were defined. The key observation was that one can define bihamiltonian pencils from a given Poisson tensor using scaling arguments, and hence apply a recipe used in [11].Our derivation of the equation resembles the one of [18]; however, on the one hand we will take advantage of the fact that the phase space is the dual of a Lie algebra, and, on the other hand, we will require that the hydrodynamical limit of the resulting equations be "substantially" the same as that of the classical equation we are starting from. As it will be briefly sketched in Section 3, the latter are the well known AKNS equations, that is, a complex form of the Nonlinear Schrödinger equations.In the core of the paper we will study both formal and concrete properties of the hierarchy (1.1), which we call CH2 hierarchy. At first we will characterize it from the bihamiltonian point of view, find its Lax representation, and discuss the associated negative hierarchy.

In this paper we study properties of Lax and Transfer matrices associated with quantum integrable systems. Our point of view stems from the fact that their elements satisfy special commutation properties, considered by Yu. I. Manin some twenty years ago at the beginning of Quantum Group Theory. They are the commutation properties of matrix elements of linear homomorphisms between polynomial rings; more explicitly they read: 1) elements in the same column commute; 2) commutators of the cross terms are equal:The main aim of this paper is twofold: on the one hand we observe and prove that such matrices (which we call Manin matrices for short) behave almost as well as matrices with commutative elements. Namely theorems of linear algebra (e.g., a natural definition of the determinant, the Cayley-Hamilton theorem, the Newton identities and so on and so forth) have a straightforward counterpart in the case of Manin matrices.On the other hand, we remark that such matrices are somewhat ubiquitous in the theory of quantum integrability. For instance, Manin matrices (and their q-analogs) include matrices satisfying the Yang-Baxter relation "RTT=TTR" and the so-called Cartier-Foata matrices. Also, they enter Talalaev's remarkable formulas: det(∂ z − L Gaudin (z)), det(1 − e −∂z T Y angian (z)) for the "quantum spectral curve", and appear in separation of variables problem and Capelli identities.We show that theorems of linear algebra, after being established for such matrices, have various applications to quantum integrable systems and Lie algebras, e.g in the construction of new generators in Z(U crit ( gl n )) (and, in general, in the construction of quantum conservation laws), in the Knizhnik-Zamolodchikov equation, and in the problem of Wick ordering.We also discuss applications to the separation of variables problem, new Capelli identities and the Langlands correspondence.

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We investigate global properties of the mappings entering the description of symmetries of integrable spin and vertex models, by exploiting their nature of birational transformations of projective spaces. We give an algorithmic analysis of the structure of invariants of such mappings. We discuss some characteristic conditions for their (quasi)-integrability, and in particular its links with their singularities (in the 2-plane). Finally, we describe some of their properties qua dynamical systems, making contact with Arnol'd's notion of complexity, and exemplify remarkable behaviours. PAR-LPTHE92/26 To appear in Communications in Mathematical Physics * Supported in part by Ministère de la Recherche et de la Technologie I I J J J J