Left-hand panels: PM distribution of the blue and red M31 samples as a function of G_BP − G_RP. The data points in the left (right) panel show the weighted average differences between the observed PMs, ${\mu }_{\alpha ,\mathrm{obs}}^{\ast }$ (μ_δ, obs), from Gaia DR3 and the predicted rotational components ${\mu }_{\alpha ,\mathrm{RC}}^{\ast }$ (${\mu }_{\delta ,\mathrm{RC}}$), derived from the RC in Sect. 3.1.1. Weights are proportional to 1/σ_α*² or 1/σ_δ², and the errors, also weighted by uncertainty, are indicated accordingly. Right-hand panels: Similar to the left-hand panels, but for M33.