Rotation of halo populations in the Milky Way and M31

We search for signs of rotation in the subsystems of the Milky Way and M31 that are defined by their satellite galaxies, their globular cluster populations and their Blue Horizontal Branch (BHB) stars. A set of simple distribution functions is introduced to describe anisotropic and rotating stellar populations embedded in dark haloes of approximate Navarro-Frenk-White form. The BHB stars in the Milky Way halo exhibit a dichotomy between a prograde-rotating, comparatively metal-rich component ([Fe/H] > -2) and a retrograde-rotating, comparatively metal-poor ([Fe/H] < -2) component. The prograde metal-rich population may be associated with the accretion of a massive satellite (similar to 10(9) M-circle dot). The metal-poor population may characterize the primordial stellar halo and the net retrograde rotation could then reflect an underestimate in our adopted local standard of rest circular velocity Theta(0). If Theta(0) is approximate to 240 km s(-1), then the metal-poor component has no rotation and there is a net prograde rotation signal of approximate to 45 km s(-1) in the metal-rich component. There is reasonable evidence that the Milky Way globular cluster and satellite galaxy systems are rotating with < v(phi)> approximate to 50 and 40 km s(-1), respectively. Furthermore, a stronger signal is found for the satellite galaxies when the angular momentum vector of the satellites is inclined with respect to the normal of the disc. The dwarf spheroidal satellites of M31 exhibit prograde rotation relative to the M31 disc with < v(phi)> approximate to 40 km s(-1). We postulate that this group of dwarf spheroidals may share a common origin. We also find strong evidence for systemic rotation in the globular clusters of M31 particularly for the most metal-rich.