Inside the algebra LT Z (R) of Z × Z-matrices with coefficients from a commutative C-algebra R that have only a finite number of nonzero diagonals above the central diagonal, we consider a deformation of a commutative Lie algebra C sh (C) of finite band skew hermitian matrices that is different from the Lie subalgebras that were deformed at the discrete KP hierarchy and its strict version. The evolution equations that the deformed generators of C sh (C) have to satisfy are determined by the decomposition of LT Z (R) in the direct sum of an algebra of lower triangular matrices and the finite band skew hermitian matrices. This yields then the C sh (C)-hierarchy. We show that the projections of a solution satisfy zero curvature relations and that it suffices to solve an associated Cauchy problem. Solutions of this type can be obtained by finding appropriate vectors in the LT Z (R)-module of oscillating matrices, the so-called wave matrices, that satisfy a set of equations in the oscillating matrices, called the linearization of the C sh (C)-hierarchy. Finally, a Hilbert Lie group will be introduced from which wave matrices for the C sh (C)-hierarchy are constructed. There is a real analogue of the C sh (C)hierarchy called the Cas(R)-hierarchy. It consists of a deformation of a commutative Lie algebra Cas(R) of anti-symmetric matrices. We will properly introduce it here too on the way and mention everywhere the corresponding result for this hierarchy, but we leave its proofs mostly to the reader.

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A new metric in the space clos(X) of all closed subsets of a metric space X is proposed. This metric, unlike the generalized Hausdorff metric, takes finite values only, and the convergence of a sequence of closed sets H i , i = 1, 2, . . . , with respect to this metric is equivalent to the convergence (in the sense of Hausdorff ) for any r ≥ 0 of the unions of H i with a closed 'exterior ball' of radius r. Using this metric allows one to investigate multi-valued maps that have images in clos(X) and are not continuous in the Hausdorff metric. In the work, the necessary and sufficient conditions for a multi-valued map to be continuous and Lipschitz with respect to the metric presented are studied, a connection of these properties with their analogues in the Hausdorff metric is derived, and a generalization of the Nadler fixed point theorem is obtained. MSC: 47H04; 47H10; 54E35

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