Dark matter halo play a key role in many models of galaxies among which parametrized models of galaxy-halo relationship have a wide range of applications in astrophysics and cosmology. This paper investigates the in-plane and out-of-plane motion of a restricted mass in presence of dark matter halo, which is characterized by a dimensionless parameter \(\gamma \in [0,3)\) under the frame of restricted three body problem. First, the equations of motion are obtained and positions of all equilibrium points are computed followed by stability test is performed. It is observed that all collinear equilibrium points \(L_1\) , \(L_2\) , \(L_3\) and out-of-plane equilibrium points \(L_{6,7}\) are unstable and change their individual positions within the range of \(\gamma \in [0,\,3)\) and \(\mu \in (0,\,1/2]\) , while non-collinear equilibrium points \(L_{4,5}\) are stable in linear sense within the range of stability as \(\gamma \in [0,\,3)\) and \(\mu \in (0,\,\mu _c]\) but varied their positions with \(\gamma\) . Additionally, the orbits in the vicinity of \(L_4\) point are determined and it is found that amplitude, time-period and orientation change with respect to the dark matter halo parameter \(\gamma\) . Next, to visualize the prohibited zones for the motion of restricted mass, the zero-velocity curves are estimated at different values of \(\gamma\) and \(\mu\) for both, in-plane and out-of-plane motion and it is seen that these are expanding with mass ratio \(\mu\) and dark matter halo parameter \(\gamma\) . The findings of this study are not only useful to enhance the mission design for space exploration but also helps to describe the perturbed motion in the different galactic systems.
