We present a three-parameter, self-similar family of steady, axisymmetric, nonrelativistic solutions that unifies the morphology, kinematics, and viscous transport of accretion–ejection flows. The triplet \((\alpha ,\beta ,\gamma )\) governs the radial power-law indices of angular velocity, density, and kinematic viscosity, respectively. In the inviscid limit, the geometric index \(\alpha\) continuously organizes the flow topology—from flared, toroidal envelopes ( \(\alpha <2\) ) through the cylindrical limit ( \(\alpha =2\) ) to collimated, jet-like funnels ( \(\alpha>2\) )—while the stratification index \(\beta\) controls mass loading and helical pitch. Introducing a scale-free viscosity \(\nu (r)\propto r^{\gamma }\) preserves separability and yields an analytic viscous correction \(\propto r^{\gamma -1}\) to the meridional velocities, with amplitude set by a coupling \(V_{\gamma }\) . This framework provides closed-form expressions for velocity fields, streamlines, and stream surfaces, enabling quantitative morphology diagnostics such as the opening-angle profile \(\psi (\theta )\) and contour-based RMSE for direct comparison with simulations or observations. The resulting \((\alpha ,\beta ,\gamma )\) atlas defines a transparent analytic baseline for global HD/GRMHD models, clarifies how viscosity tilts self-similar stream surfaces, and offers benchmark solutions for reduced or physics-informed neural network surrogates.
