The characteristic system of point correspondences

The characteristic system (of point correspondence between projective spaces $P_n$, $P'_n$) defined in [1] as the linear system of cubic cones $a_{ij}\Phi^{ij}=0$ ($\Phi^{ij}=\omega^i\Omega^j-\omega^j\Omega^i$ — fundamental cones $\omega^i$ — homogeneous coordinates of line at $М\in P_n$, $\Omega^i$ — quadric forms in $\omega^i$) is examined. For some classes of characteristic systems the problem of the uniqueness of the fundamental cones is solved. The expression of the other system of fundamental cones $\tilde\Phi^{ij}$ is given under certain complementary conditions on the cones $\Phi^{ij}$.