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
$$ (\widetilde{A}_1f)(x,y)=i\int_x^af(t,y)\,dt,\qquad(\widetilde{A}_2f)(x,y)=i\int_y^bf(x,s)\,ds $$
in $L_{\Omega_L}^2$, where $\Omega_L$ is the compact set in $\mathbb{R}_+^2$ bounded by the lines $x=a$ and $y=b$ and by a decreasing smooth curve $L=\{(x,p(x)):p(x)\in C_{[0,a]}^1,\,p(0)=b,\,p(a)=0\}$.