Is there a way to construct a unit vector field $V(\textbf{x})=(V_x,V_y,\sqrt{1-V_x^2-V_y^2})$ in $\mathbb{R}^3$ knowing its divergence $\nabla \cdot V$ and Darboux vector $\omega=\kappa(\textbf{x}) (V(\textbf{x})\times V'(\textbf{x})) + \tau(\textbf{x}) V(\textbf{x})$ at every point, where $\kappa(\textbf{x}),\tau(\textbf{x})$ are also known?
How could I go about proving either?
So far I have tried brute force through substitution a component of the divergence into the other scalars $\kappa(\textbf{x}), \tau(\textbf{x})$, but I end up with a differential equation which is very complicated. Is there a known approach for the construction of vector fields from their "internal" scalars?