Inversion of integral operators with kernels discontinuous on the diagonal

Conditions implying the invertibility of the integral operator
$$ Af(x)=\int_0^1A(x,t)f(t)\,dt $$
with kernel $A(x,t)$ having discontinuities of the first kind at the points $t=x$ and $t=1-x$ are found. We give explicit inversion formulas as well as applications to the problem of finding the square roots of the operator $y''(x)$ with arbitrary boundary conditions and the problem of expansion with respect to eigenfunctions.