  In this paper, we reduce the general linear integral equation of the third
kind in $L^2(Y,\mu)$, with largely arbitrary kernel and coefficient, to an
equivalent integral equation either of the second kind or of the first kind in
$L^2(\mathbb{R})$, with the kernel being the linear pencil of bounded
infinitely differentiable bi-Carleman kernels expandable in absolutely and
uniformly convergent bilinear series. The reduction is done by using unitary
equivalence transformations.
