In this work, we address the stability of equilibrium points for particle motion in an important class of flows for which a consistent Lagrangian equation of motion for the particle can be derived and solved exactly. This class of flows includes all nonuniform, steady solenoidal flows that are characterized by constant vorticity and constant strain rate. We demonstrate that equilibrium points for particle motion exist for such flow fields and that, under some specific conditions, these equilibrium points are stable and therefore represent the observable steady-state solution of the Lagrangian equation of motion for the particle. We show that there exists an exact solution procedure for the Maxey-Riley equation of motion in all linear solenoidal flows. We also show that for two-dimensional linear solenoidal velocity fields characterized by closed streamlines, the stability of the equilibrium point for the particle motion is determined by the relative density of the particle with respect to the fluid: heavy particles produce unstable equilibrium points and light particles produce stable equilibrium points. For flows that are not characterized by closed streamlines, the equilibrium points for the particle motion for both heavy and light particles are unstable. Finally, we completely characterize three-dimensional linear solenoidal velocity fields that are solutions of the Navier-Stokes equations and establish the necessary and sufficient conditions for the stability of particle equilibrium points in such flows.