We present analytical and numerical solutions which describe a hot, viscous, two-temperature accretion flow onto a rotating neutron star or any other rotating compact star with a surface. We assume Coulomb coupling between the protons and electrons, and free-free cooling from the electrons. Outside a thin boundary layer, where the accretion flow meets the star, we show that there is an extended settling region which is well-described by two self-similar solutions: (i) a two-temperature solution which is valid in an inner zoner⩽102.5(ris in Schwarzchild units), and (ii) a one-temperature solution at larger radii. In both zones,&rgr;∝r−2,&OHgr;∝r−3/2,v∝r0,Tp∝r−1;in the two-temperature zone,Te∝r−1/2.The luminosity of the settling zone arises from the rotational energy of the star as the star is braked by viscosity. Hence the luminosity and the flow parameters (density, temperature, angular velocity) are independent ofM. The settling solution described here is not advection-dominated, and is thus different from the self-similar ADAF found around black holes. When the spin of the star is small enough, however, the present solution transforms smoothly to a (settling) ADAF. We carried out a stability analysis of the settling flow. The flow is convectively and viscously stable and is unlikely to have strong winds or outflows. Unlike another cooling-dominated system—the SLE disk,—the settling flow is thermally stable provided that thermal conduction is taken into account. This strong saturated-like thermoconduction does not change the structure of the flow. ©2001 American Institute of Physics.