The celebrated Poincaré (1910) steady solution, the uniform vorticity flow in a precessing spheroid, which is realized in the inviscid interior region outside the boundary layer in the strong spin and weak precession limit, is derived, as a unique solution analytically without any prior assumption on the spatial structure, and its singular behavior for a sphere is resolved. Assuming that the spin and precession axes are respectively parallel and perpendicular to the symmetry axis of the spheroid, we denote the spin and precession angular velocities by and respectively, and define the Reynolds number Re = and the Poincaré number , where a is the equatorial radius and ν is the kinematic viscosity of fluid. An ellipsoidal coordinate system is introduced, which makes it possible to express the solution explicitly in a separable form. It is shown that only the uniform vorticity flow can develop as a steady flow in the inviscid limit. The vorticity, relative to the spinning spheroid, is in magnitude and points to or against the z-axis for an oblate (c < 1) or a prolate (c > 1) spheroid. This expression of vorticity magnitude diverges to infinity for a sphere (c = 1), where the inviscid solution is degenerate. This apparent singular behavior of the vorticity is resolved by introducing viscosity and analyzing the flow by invoking the torque balance equation for a spheroid close to a sphere in the limit , where . It is found that the vorticity vector changes the orientation smoothly by 180° over and that the parallel and the perpendicular (to the spin axis) components of vorticity is expressed, up to the nontrivial leading order, as the twice of −0.7876Po2/[(2.620δ)2 + (0.2585δ + 1 − c)2] and , respectively. The latter is included in Busse ()'s formula, whereas the former is new. An analysis of the same flow as the present paper from a different approach was performed recently by Zhang et al (). Unfortunately, however, an inconsistency is found in their results.