Singular integral equations of nonclassical type on a piecewise smooth curve

We consider singular integral operators on a piecewise smooth curve in weighted Lebesgue and Hölder spaces with piecewise continuous matrix coefficients. In contrast to the classical case, these operators, in addition to the singular Cauchy operator, also contain non-compact integral operators of a special form which are defined by the kernel that is approximately homogeneous of degree $-1$ with respect to distances to the nodes of the curve. Similar operators arise in many applications. A criterion for the Fredholm property of these operators is obtained and a formula for their index is given.