Results for the double ring (1, 2). We consider the parameters a = 4 kpc, σ = 24.80 km s^-1, , κ = 1.2, ρ = 0.12 × 10^-13 kg m^-3, m_a = M_⊙, v_typ = 0.8σ, b_m = 4 kpc, δz = 3 × 10^-9 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 density profile obtained from the numerical integration of Eq. (6) (full circles). In the region with the highest density the difference is less than 1.7%. 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 ≈ 2.32, respectively. Graphs e) and f) show the spatial distribution function and the level curves for the case v_ϕ = 0, respectively.