Geodesic
On~the complete integrability of~a~nonlinear system of~partial differential equations in~two-dimensional space
Teoretičeskaâ i matematičeskaâ fizika, Tome 42 (1980) no. 3, pp. 343-349

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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. 
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     author = {A. N. Leznov},
     title = {On~the complete integrability of~a~nonlinear system of~partial differential equations in~two-dimensional space},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {343--349},
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     number = {3},
     year = {1980},
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A. N. Leznov. On~the complete integrability of~a~nonlinear system of~partial differential equations in~two-dimensional space. Teoretičeskaâ i matematičeskaâ fizika, Tome 42 (1980) no. 3, pp. 343-349. http://geodesic.mathdoc.fr/item/TMF_1980_42_3_a4/