Geodesic
Darboux system: Liouville reduction and an explicit solution
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Topology and physics, Tome 302 (2018), pp. 268-286

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A class of solutions to a Darboux system in $\mathbb R^3$ is introduced that satisfy the factorization condition for an auxiliary second-order linear problem. It is shown that this reduction provides the (local) solvability of the Darboux system, and an explicit solution is given to this problem for two types of dependent variables. Explicit formulas for the Lamé coefficients and solutions to the associated linear problem are constructed. It is shown that the reduction, known in the literature, to a weakly nonlinear system is a particular case of the approach proposed. 
@article{TM_2018_302_a11,
     author = {R. Ch. Kulaev and A. K. Pogrebkov and A. B. Shabat},
     title = {Darboux system: {Liouville} reduction and an explicit solution},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {268--286},
     publisher = {mathdoc},
     volume = {302},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2018_302_a11/}
}
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%D 2018
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R. Ch. Kulaev; A. K. Pogrebkov; A. B. Shabat. Darboux system: Liouville reduction and an explicit solution. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Topology and physics, Tome 302 (2018), pp. 268-286. http://geodesic.mathdoc.fr/item/TM_2018_302_a11/