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We first stress that any spherically symmetric galactic model whoseintegrated mass profile $M (r) \to 0$ as $r \to 0$ is physically consistentclose to the centre only provided that the circular velocity $v_c (r) \to 0$and the gravitational field $g (r) \to 0$ as $r \to 0$. Next, we apply such astatement to a broad class of five-parameter spherical galactic models, whichincludes most of those used in astrophysics and cosmology. In particular, wediscover that the Jaffe and Hernquist models can only be trusted for $r \gtrsim0.2 \, R_e$, while the NFW model cannot describe the central region either ofregular galaxy clusters or of pure dark matter halos, thereby failing topredict any central cusp.

New strong constraints on the central behaviour of spherical galactic models -- No NFW cusp