Results for the ring (1,1). We consider the parameters a = 4 kpc, σ = 48.87 km s^-1, ℳ^(1,1) = 10⁹ M_⊙, ρ = 1.35 × 10^-13 kg m^-3, m_a = M_⊙, v_typ = 0.8σ, b_m = 4 kpc, δz = 6 × 10^-10 kpc, σ_v = 0.8σ, and r_max = 4a. In graph a) the mass surface density profile of the exact solution (solid line) is compared with the mass surface profile obtained from the numerical integration of Eq. (6) (full circles). In the region with the highest density, the difference between them is less than 0.8%. Graph b) shows the values of ⟨ v² ⟩ /σ² within the ring. Graphs c) and d) show the velocity distribution function and the level curves for case r/a ≈ 0.72, respectively. Graphs e) and f) show the spatial distribution function and the level curves for case v_ϕ = 0, respectively.