In the strip $[0;T] × \mathbb{R}^p$,  we investigate a problem with integral-boundary conditions in time coordinate $t ∈ [0;T]$  for generalized system of Lamé equations of dynamic elasticity theory, in the class of almost periodical functions in spatial variables $x_1,...,x_p$.  We found a uniqueness criterion, and necessary, necessary and sufficient, and sufficient existence conditions for the solution of this problem. To solve the problem of small denominators arising while constructing a solution of the problem, we use the metrical approach.

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