Darboux system and separation of variables in the Goursat problem for a third order equation in $\mathbb {R}^3$

We construct a reduction of the three-dimensional Darboux system for the Christoffel symbols, which describes conjugate curvilinear coordinate systems. The reduction is determined by one additional algebraic condition on the Christoffel symbols. It is shown that the corresponding class of solutions of the Darboux system is parametrized by six functions of one variable (two for each of three independent variables). Explicit formulas for Darboux system solutions are given. For the case when Christoffel symbols are constants, the linear system associated with the Darboux system is studied. In this formulation, this system is reduced to the three-dimensional Goursat problem for a third-order equation with data on the coordinate planes. It is shown that the solution to the Goursat problem allows the separation of variables and is determined by its values on the coordinate lines.