  We propose two generalizations of the dark photon model which predict the
suppressed elastic dark matter nuclei cross section in comparison with the
corresponding prediction of the dark photon model. In the first model the main
difference from dark photon model is that the mixing parameter $\epsilon$ is
nonlocal formfactor $\epsilon(q^2)=
\frac{q^2}{\Lambda^2}V(\frac{q^2}{\Lambda^2}) $ depending on the square of the
momentum transfer $q^2$. Here $V(\frac{q^2}{\Lambda^2})$ is an entire function
of the growth $\rho \geq \frac{1}{2}$ and $\Lambda$ is nonlocal scale. In this
model our world and dark world are described by renormalizable field theories
while the communication between them is performed by nonlocal interaction. The
second model is renormalizable model where besides dark photon field $A'$
additional vector boson $Z'$ interacts with $B - L$ current. The communication
between our world and dark world is performed due to nonzero kinetic mixing
between $Z'$ and $A'$ fields. The predictions of the models for the search for
dark matter at the accelerators don't contain additional suppression factors.
