2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/178373We introduce exact analytical solutions of the steady-state hydrodynamic equations of scale-free, self-gravitating gaseous disks with flat rotation curves. We express the velocity field in terms of a stream function and obtain a third-order ordinary differential equation (ODE) for the angular part of the stream function. We present the closed-form solutions of the obtained ODE and construct hydrodynamical counterparts of the power-law and elliptic disks, for which self-consistent stellar dynamical models are known. We show that the kinematics of the Large Magellanic Cloud can well be explained by our findings for scale-free elliptic disks.AAS preprint format, 21 pages, 8 figures, accepted for publication in The Astrophysical JournalAstrophysicstext