  Considered a unified field theory approach describing matter and space
(metric tensor) by means of a 3-index tensor $P^{i}_{jk}$. It is shown that if
the Lagrangian has partial U(1) gauge (semi-gauge) of the type
$P^{i}_{jk}->P^{i}_{jk}+a \delta{^i_j} \phi{_{,k}}+b \delta{^i_k}
\phi{_{,j}}+cg_{jk}g^{mi} \phi{_{,m}}$ where a,b,c are constants, then the
Euler equations of motion contain the covariant low of conservation in the form
$J^{k}_{;k}=0$.
