  In this text matrix Volterra integral equation of the first kind is
addressed. It is assumed that kernels of the equation have jump discontinuities
on non-intersecting curves. Such equations appear in the theory of evolving
dynamic systems. Differentiation of such equations with jump discontinue
kernels yields the new class of the Volterra integral equations with
functionally perturbed argument. The algorithm for construction of the
logarithmic power asymptotics of the desired continuous solutions is proposed.
The theorem of existance of the parametric families of solutions is proved.
Finally the sufficient conditions for existence and uniqueness of continuous
solution are derived.
