I was seeking for the solution of following partial differential equation for two unknowns $\vec{u}(s,t), \vec{w}(s,t)$ $$\partial_t \vec{u} = \partial_s \vec{w} - [\vec{w} \times \vec{u}].$$ Using computed-based Lie symmetries method I achieved to get its general solution $$\vec{u} = \partial_s \vec{a} + \frac{a - \sin a}{a^3} (\vec{a} (\vec{a} \cdot \partial_s \vec{a}) - a^2 \partial_s \vec{a}) - \frac{1 - \cos a}{a^2} [\vec{a} \times \partial_s \vec{a}],$$ $$\vec{w} = \partial_t \vec{a} + \frac{a - \sin a}{a^3} (\vec{a} (\vec{a} \cdot \partial_t \vec{a}) - a^2 \partial_t \vec{a}) - \frac{1 - \cos a}{a^2} [\vec{a} \times \partial_t \vec{a}].$$ Actually, it was obtained by purely algebraic methods, not involving its intrinsic geometric nature. It would be a great pleasure for me, if it could be derived by methods of differential geometry.

Do you have any ideas?