= Solution
With the convention in the question, the constant <velocity-anisotropy parameter> is $\beta=1-\sigma_t^2/\sigma_r^2$. Since $V_c^2=r\,d\Phi/dr$, the <Spherical Jeans equation> becomes
$$
\frac{d(\nu\sigma_r^2)}{dr}
+\frac{2\beta}{r}\nu\sigma_r^2
=-\frac{\nu V_c^2}{r}.
$$
For $\nu=\nu_0r^{-\alpha}$ and $V_c^2=V_0^2r^{2\gamma}$, its general <integrating factor> solution is
$$
\boxed{\sigma_r^2(r)=
\frac{V_c^2(r)}{\alpha-2\beta-2\gamma}
+Cr^{\alpha-2\beta}.}
$$
The homogeneous term represents a boundary pressure. For an extended scale-free system the physical boundary condition normally removes it, leaving
$$
\boxed{\sigma_r^2=\frac{V_c^2}{\alpha-2\beta-2\gamma}.}
$$
Positivity requires $\alpha-2\beta-2\gamma>0$. A complete physical model must also have a nonnegative <galactic distribution function> and sensible inner and outer boundary behaviour; for example, strong radial anisotropy is restricted by density-slope--anisotropy inequalities.
Observationally, the tracer density can be estimated from star counts only after correcting distances, extinction, survey selection, and incompleteness. Spectroscopy supplies mainly line-of-sight velocities; proper motions add transverse information but become less precise for distant halo stars. The equation shows the <mass--anisotropy--density degeneracy> directly: the same measured $\sigma_r$ can result from a larger $V_c$, a steeper tracer slope $\alpha$, or a different $\beta$. Even globally constant power laws therefore do not determine the galactic mass profile unless some of these quantities are independently constrained.
If $\alpha$, $\beta$, or $\gamma$ changes near a break radius, the solution at one radius also depends on the outer boundary integral. A break in observed dispersion may be attributed to a mass-profile feature, a tracer-density break, or a change in orbital anisotropy. Separate tracer populations, full three-dimensional velocities, higher velocity moments, and measurements over a wide radial range help break this degeneracy.
More flexible alternatives model a nonnegative solution of the <Collisionless Boltzmann equation> itself. An <action-based galactic distribution function> gives an analytic or parametrized $f(\mathbf J)$; a <Schwarzschild orbit-superposition model> assigns nonnegative weights to an orbit library; and a <made-to-measure stellar-dynamical model> adjusts particle weights to reproduce observations. These methods retain more phase-space information than Jeans moments, although their flexibility introduces model choices and regularization.
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