On the complete integrability of a nonlinear system of partial differential equations in two-dimensional space

For the system of nonlinear partial differential equations of the form
$$ \rho_{\alpha,z\bar z}=\sum_{\beta=1}^rk_{\alpha\beta}\exp\rho_\beta $$
where $k$ is the Cartan matrix of a certain semisimple algebra of the rank $r$, the complete integrability is demonstrated by means of constructing the complete solutions in an explicit form which depend on $2r$ arbitrary functions.