  We investigate, using the spherical Jeans equation, self-gravitating
dynamical equilibria satisfying a relation rho/sigma_r^3 propto r^-alpha, which
holds for simulated dark-matter haloes over their whole resolved radial range.
Considering first the case of velocity isotropy, we find that this problem has
only one solution with realistic density profile, which occurs only for a
critical value of alpha_crit = 35/18 ~= 1.94, which is consistent with the
empirical value of 1.9+/-0.05. We extend our analysis in two ways: first we
introduce a parameter epsilon to allow for a more general relation
rho/\sigma_r^epsilon propto r^-alpha; second we consider velocity anisotropy.
If we assume beta(r) := 1- sigma_theta^2 / sigma_r^2 to be linearly related to
the logarithmic density slope gamma(r) := -dln(rho)/dln(r), which is in
agreement with simulations, the problem remains analytically tractable and is
equivalent to the simpler isotropic case: there exists only one physical
solution, which occurs at a critical alpha value. Remarkably, this value of
alpha and the density and velocity-dispersion profiles depend only on epsilon
and the value beta_0 := beta(r=0), but not on the slope of the linear
beta-gamma relation. For epsilon=3, alpha_crit = 35/18 - 2beta_0/9 and the
resulting density profile is fully analytic (as are the velocity dispersion and
circular speed) with an inner cusp rho propto r^{-(7+10beta_0)/9} and a very
smooth transition to a steeper outer power-law asymptote. These models are in
excellent agreement with the density, velocity-dispersion and anisotropy
profiles of simulated dark-matter haloes over their full resolved radial range.
If epsilon=3 is a universal constant, some scatter in beta_0 ~= 0 may account
for some diversity in the density profiles. (ABRIDGED)
