We investigate, using the spherical Jeans equation, self-gravitating dynamical equilibria satisfying a relation ρ/σ3r∝r−α, 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 for which the density profile is not truncated or otherwise unrealistic. This solution occurs only for a critical value of Graphic, which is consistent with the empirical value of 1.9 ± 0.05. We extend our analysis in two ways: first, we introduce a parameter ε to allow for a more general relation ρ/σεr∝r−α; and secondly, we consider velocity anisotropy parametrized by Binney's β(r) = 1 −σ2θ/σ2r. If we assume β to be linearly related to the logarithmic density slope γ(r) =−(d ln ρ/d ln 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 a value. Remarkably, this value of α, and the density and velocity-dispersion profiles, depend only on ε and the value β0=β(r= 0), but not on the value β∞=β(r→∞) (or, equivalently, the slope dβ/dγ of the adopted linear β-γ relation). For ε= 3, αcrit= 35/18 − 2β0/9 and the resulting density profile is fully analytic (as are the velocity dispersion and circular speed) with an inner cusp Graphic 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 ε= 3 is a universal constant, some scatter in β0≈ 0 may account for some diversity in the density profiles, provided a relation Graphic always holds
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