On a certain class of commuting systems of linear operators

In this paper we describe the class of commuting pairs of bounded linear operators {A ₁,A ₂} acting on a Hilbert space H which are unitarily equivalent to the system of integrations over independent variables $$ (\tilde A_1 f)(x,y) = i\int_x^a {f(t,y)} dt, (\tilde A_2 f)(x,y) = i\int_y^b {f(x,s)} ds $$ in $ L_{\Omega _L }^2 $ , where ?? _L is the compact set in ? ₊ ² bounded by the lines x = a and y = b and by a decreasing smooth curve L = {((x, p(x)): p(x) ?? C _0,a] ¹ , p(0) = b, p(a) = 0}.