  Measurements of the kinematic distributions of $J/\psi$ mesons produced in
$p-$C, $p-$Ti and $p-$W collisions at $\sqrt{s}=41.6 \mathrm{GeV}$ in the
Feynman-$x$ region $-0.34 < x_{F} < 0.14$ and for transverse momentum up to
$p_T = 5.4 \mathrm{GeV}/c$ are presented. The $x_F$ and $p_T$ dependencies of
the nuclear suppression parameter, $\alpha$, are also given. The results are
based on $2.4 \cdot 10^{5}$ $J/\psi$ mesons in both the $e^+ e^-$ and
$\mu^{+}\mu^{-}$ decay channels. The data have been collected by the HERA-B
experiment at the HERA proton ring of the DESY laboratory. The measurement
explores the negative region of $x_{F}$ for the first time. The average value
of $\alpha$ in the measured $x_{F}$ region is $0.981 \pm 0.015$. The data
suggest that the strong nuclear suppression of $J/\psi$ production previously
observed at high $x_F$ turns into an enhancement at negative $x_F$.
