Explicit solution of the inverse kinematic problem in the non-Herglotz case

The inverse kinematic problem is solved in the half space $R_+^{\nu+1}=\{(x,z)\mid z\geqslant0,\ x\in R^\nu\}$, $\nu\geqslant1$ under the assumption that the index of refraction can be represented in the form
$$ n^2(x,z)=k^2(z)+\sum^\nu_{j=1}\Phi^2_j(x_j),\quad n_z<0. $$
The solution obtained is a generalization of the Herglotz–Wiechert formula. A formula is presented for the solution of the inverse kinematic problem in the general case of separation of variables in the eikonal equation.