By the methods of the operators semigroups theory a degenerate linear evolution equation with memory in a Banach space is reduced to a system of two equations. One of them is resolved with respect to the derivative, another has a nilpotent operator at the derivative. A problem with a given history for the first of the equations with memory is brought to the Cauchy problem for stationary equations system in a wider space. It allowed to obtain conditions of the unique solution existence for the problem, including solutions with a greater smoothness, by the methods of the classical operators semigroups theory. Thus unique solvability of the problem with a given history for a degenerate linear evolution equation with memory was researched with using some restrictions for the kernel of the memory integral operator. Besides, an analogous problemь with generalized Showalter – Sidorov type condition on the history of the system was studied. General results were used for investigation of an initial boundary value problem for the linearized Oskolkov integro-differential system of equations, descibing the dynamics of the high order Kelvin – Voight fluid.

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