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 BHB stars .
A set of simple distribution functions are 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 ( [ \mathrm { Fe / H } ] > -2 ) and a retrograde rotating , comparatively metal-poor ( [ \mathrm { Fe / H } ] < -2 ) component .
The prograde metal-rich population may be associated with the accretion of a massive satellite ( \sim 10 ^ { 9 } M _ { \odot } ) .
The metal-poor population may characterise 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 \approx 240 \mathrm { kms ^ { -1 } } then the metal-poor component has no rotation and there is a net prograde rotation signal of \approx 45 \mathrm { kms ^ { -1 } } in the metal-rich component .
There is reasonable evidence that the Milky Way globular cluster and satellite galaxy systems are rotating with \langle v _ { \phi } \rangle \approx 50 \mathrm { kms ^ { -1 } } and \langle v _ { \phi } \rangle \approx 40 \mathrm { kms ^ { -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 \langle v _ { \phi } \rangle \approx 40 \mathrm { kms ^ { -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 .