Results for the homogeneous ring model. We consider the following parameters a = 10 kpc, b = 12 kpc, σ = 36.95 km s^-1, ℳ_R = 10⁹M_⊙, ρ = 0.24 × 10^-13 kg m^-3, m_a = M_⊙, v_typ = 0.8σ, b_m = 2 kpc, δz = 2 × 10^-8 kpc, and σ_v = 0.9σ. In graph a) the mass surface density profile of the exact solution (solid line) is compared with the mass surface density profile obtained from the numerical integration of Eq. (6) (full circles). The highest difference between them is less than 2.2%. 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 the case r/a ≈ 1.09, respectively. Graphs e) and f) show the spatial distribution function and the level curves for the case v_r = 0, respectively.