First of all note that 8-dinensional Finsler space (t,x,y,z,t^*,x^*,y^*,z^*) preserving the metric form
\begin{equation}
S^2 = tt^*-xx^*-yy^*-zz^*,
\end{equation}
actually presents doubled of the Minkowski space.
Then the solution with one-dimensional feature localized on the world line...