We show that eigenvalues of the family of Baxter Q-operators for supersymmetric integrable spin chains constructed with the gl(K|M )-invariant R-matrix obey the Hirota bilinear difference equation. The nested Bethe ansatz for super spin chains, with any choice of simple root system, is then treated as a discrete dynamical system for zeros of polynomial solutions to the Hirota equation. Our basic tool is a chain of Bäcklund transformations for the Hirota equation connecting quantum transfer matrices. This approach also provides a systematic way to derive the complete set of generalized Baxter equations for super spin chains.

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We formulate a conjecture for the three different Lax operators that describe the bosonic sectors of the three possible N = 2 supersymmetric integrable hierarchies with N = 2 super W n second hamiltonian structure. We check this conjecture in the simplest cases, then we verify it in general in one of the three possible supersymmetric extensions. To this end we construct the N = 2 supersymmetric extensions of the Generalized Non-Linear Schrödinger hierarchy by exhibiting the corresponding super Lax operator. To find the correct hamiltonians we are led to a new definition of super-residues for degenerate N=2 supersymmetric pseudodifferential operators. We have found a new non-polinomial Miura-like realization for N = 2 superconformal algebra in terms of two bosonic chiral-anti-chiral free superfields.

We show that the Zamolodchikov's and Polyakov-Bershadsky nonlinear algebras W 3 and W (2) 3 can be embedded as subalgebras into some linear algebras with finite set of currents. Using these linear algebras we find new field realizations of W (2) 3 and W 3 which could be a starting point for constructing new versions of W -string theories. We also reveal a number of hidden relationships between W 3 and W (2) 3 . We conjecture that similar linear algebras can exist for other W -algebras as well.

We show that the well known N = 1 NLS equation possesses N = 2 supersymmetry and thus it is actually the N = 2 NLS equation. This supersymmetry is hidden in terms of the commonly used N = 1 superfields but it becomes manifest after passing to the N = 2 ones. In terms of the new defined variables the second Hamiltonian structure of the supersymmetric NLS equation coincides with the N = 2 superconformal algebra and the N = 2 NLS equation belongs to the N = 2 a = 4 KdV hierarchy. We propose the KP-like Lax operator in terms of the N = 2 superfields which reproduces all the conserved currents for the corresponding hierarchy.

An infinite class of fermionic flows of the N=(1|1) superconformal Toda lattice hierarchy is constructed and their algebraic structure is studied. We completely solve the semi-infinite N=(1|1) Toda lattice and chain hierarchies and derive their tau functions, which may be relevant for building supersymmetric matrix models. Their bosonic limit is also discussed.

The status of the new flagship project 'Nuclotron-based Ion Collider fAcility' (NICA) at the Joint Institute for Nuclear Research (JINR, Dubna), the NICA main parameters and physics program are briefly presented. The generation of intense heavy ion and polarized light nuclear NICA beams is aimed at searching for the phase transitions, mixed phase and critical phenomena in nuclear matter and the investigation of polarization phenomena at the collision energies up to √ s NN = 9 GeV.

A manifestly N = 2 supersymmetric coset formalism is introduced to describe integrable hierarchies. It is applied to analyze the super-NLS equation. It possesses an N = 2 symmetry since it can be obtained from a manifest N = 2 coset algebra construction. A remarkable result is here discussed: the existence of a Bäcklund transformation which connects the super-NLS equation to the equations belonging to the integrable hierarchy of one particular (the a = 4) N = 2 super-KdV equation. N = 2 scalar Lax pair operators are introduced for both super-KdV and super-NLS. Submitted to Physics Letters AApril 1995 JINR E2-95-185 DFPD 95-TH-24 hep-th/9504138