On some special solutions of Eisenhart equation

In this note we study a $6$-dimensional pseudo-Riemannian space $V^6(g_{ij})$ with the signature $[++----]$, which admits projective motions, i.e., continuous transformation groups preserving geodesics. A general method of determining pseudo-Riemannian spaces admitting some nonhomothetic projective group $G_r$ was developed by A. V. Aminova. A. V. Aminova classified all Lorentzian manifolds of dimension $\geq3$ admitting nonhomothetic projective or affine infinitesimal transformations. The problem of classification is not solved for pseudo-Riemannian spaces with arbitrary signature.
In order to find a pseudo-Riemannian space admitting a nonhomothetic infinitesimal projective transformation, one has to integrate the Eisenhart equation
$$h_{ij,k}=2g_{ij}\varphi_{,k}+g_{ik}\varphi_{,j}+g_{jk}\varphi_{,i}.$$

Pseudo-Riemannian manifolds for which there exist nontrivial solutions $h_{ij}\ne cg_{ij}$ to the Eisenhart equation are called $h$-spaces. It is known that the problem of describing such spaces depends on the type of an $h$-space, i.e., on the type of the bilinear form $L_Xg_{ij}$ determined by the characteristic of the $\lambda$-matrix $(h_{ij}-\lambda g_{ij})$. The number of possible types depends on the dimension and the signature of an $h$-space.
In this work we find the metrics and determine quadratic first integrals of the corresponding geodesic lines equations for $6$-dimensional $h$-spaces of the type $[(21\ldots1)(21\ldots1)\ldots(1\ldots1)]$.