General compactly supported solution of an integral equation of the convolution type

We find the general form of solutions of the integral equation a << k(t - s)u (1)(s) ds = u (2)(t) of the convolution type for the pair of unknown functions u (1) and u (2) in the class of compactly supported continuously differentiable functions under the condition that the kernel k(t) has the Fourier transform , where and are polynomials in the exponential e (i tau x) , tau > 0, with coefficients polynomial in x. If the functions , l = 1, 2, have no common zeros, then the general solution in Fourier transforms has the form U (l) (x) = P (l) (x)R(x), l = 1, 2, where R(x) is the Fourier transform of an arbitrary compactly supported continuously differentiable function r(t).