Goursat problem for nonlinear hyperbolic systems with integrals of the first and second order

We consider the Goursat problem for one class of nonlinear hyperbolic systems of equations of the form
$$ u^i_{xy}=F^i(u, u_x, u_y),\qquad i=1,2,\quad u=(u^1,u^2), $$
with integrals of the first and second order
\begin{gather*} \omega^1(u^1,u^2,u^1_x,u^2_x),\ \omega^2(u^1,u^2,u^1_x,u^2_x,u^1_{xx},u^2_{xx}),\quad(\overline D(\omega^1)=\overline D(\omega^2)=0),\\ \overline\omega^1(u^1,u^2,u^1_y,u^2_y),\ \overline\omega^2(u^1,u^2,u^1_y,u^2_y,u^1_{yy},u^2_{yy}),\quad(D(\overline\omega^1)=D(\overline\omega^2)=0). \end{gather*}
Explicit formulas for the solutions of the Goursat problem with the data set on the characteristics
\begin{gather*} u^1(x_0,y)=\phi_1(y),\quad u^2(x_0,y)=\phi_2(y),\\ u^1(x,y_0)=\psi_1(x),\quad u^2(x,y_0)=\psi_2(x). \end{gather*}