Basis of trancendense in differential field of invariants of pseugo-Galilean group

Let $G$ be a subgroup of the group of invertible linear operators, acting in a finite-dimensional real linear space $X$. One of the problems of differential geometry is the study of necessary and sufficient conditions for $G$-equivalence of paths in $X$. In solving this problem we use methods of the theory of differential invariants describing transcendence basis of differential fields of $ G $-invariant differential rational functions. Using explicit descriptions of these bases allows to establish criteria for $G$-equivalence of paths in $X$. Such an approach has been used in solving the problem of equivalence of paths with respect to the actions of special linear, orthogonal, pseudo-orthogonal, and symplectic groups.
We give an explicit description of one of the finite bases of transcendence in the differential field of invariant differential rational functions with respect to the action of the pseudo-Galilean group $\Gamma O$. Using this basis, necessary and sufficient conditions are established for the $\Gamma O$-equivalence of paths in $X$.