= Solution
Interpret the barred <velocities> as <second moments> of a steady nonstreaming <collisionless stellar system>, and write $\sigma_r^2=\overline{v_r^2}$. In an isotropic model the three component dispersions agree, so the <Spherical Jeans equation> becomes
$$
\frac{d(\rho\sigma_r^2)}{dr}=-\frac{\rho GM(r)}{r^2}.
$$
Put $A=\rho_0r_0^k$, so $\rho=Ar^{-k}$. If this density supplies all of the <Newtonian gravity> and there is no central point <mass>, finite enclosed <mass> requires $k<3$ and
$$
M(r)=\frac{4\pi A}{3-k}r^{3-k}.
$$
Integrate to an outer radius $b$ at which the pressure is $P_b=\rho(b)\sigma_r^2(b)$. The exact Jeans-moment solution within the power-law region is
$$
\rho\sigma_r^2=P_b+\frac{4\pi GA^2}{3-k}\int_r^b s^{1-2k}\,ds,
$$
and, for $k\ne1$,
$$
\boxed{\sigma_r^2(r)=\frac{P_b}{A}r^k+\frac{2\pi GA}{(3-k)(1-k)}r^k\left(b^{2-2k}-r^{2-2k}\right).}
$$
For $k<1$ and $r\ll b$, the integral is dominated by outer radii. Therefore
$$
\boxed{\sigma_r^2\sim Cr^k,\quad C=\frac{P_b}{A}+\frac{2\pi GA}{(3-k)(1-k)}b^{2-2k}.}
$$
The coefficient cannot be determined from the local density cusp alone. Extending the power law to infinity gives a divergent pressure integral when $k<1$, so an outer scale or boundary condition is essential. For the usual shallow cusp $0<k<1$, the central <velocity dispersion> tends to zero even though the density diverges.
For $1<k<3$, inner radii dominate as $r/b\to0$; equivalently one can send $b\to\infty$ with vanishing boundary pressure to obtain
$$
\boxed{\sigma_r^2\sim\frac{2\pi G\rho_0r_0^k}{(3-k)(k-1)}r^{2-k}=\frac{V_c^2(r)}{2(k-1)}.}
$$
Thus the central <velocity dispersion> decreases for $1<k<2$, is constant at $k=2$, and diverges for $2<k<3$. At the omitted threshold $k=1$, $\sigma_r^2=(P_b/A)r+2\pi GA r\ln(b/r)$. For $k\geq3$ the assumed central density law has infinite enclosed <mass>, so the same unmodified model cannot determine a finite <velocity dispersion>. These boundary and domain restrictions are the <boundary dependence of an isotropic self-gravitating cusp>; specifying density without them does not specify a physical finite <galaxy>.
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