  We consider a set of identical mobile point-like charges (counter-ions)
confined to a domain with curved hard walls carrying a uniform fixed surface
charge density, the system as a whole being electroneutral. Three domain
geometries are considered: a pair of parallel plates, the cylinder and the
sphere. The particle system in thermal equilibrium is assumed to be described
by the nonlinear Poisson-Boltzmann theory. While the effectively 1D plates and
the 2D cylinder have already been solved, the 3D sphere problem is not
integrable. It is shown that the contact density of particles at the charged
surface is determined by a first-order Abel differential equation of the second
kind which is a counterpart of Enig's equation in the critical theory of
gravitation and combustion/explosion. This equation enables us to construct the
exact series solutions of the contact density in the regions of small and large
surface charge densities. The formalism provides, within the mean-field
Poisson-Boltzmann framework, the complete thermodynamics of counter-ions inside
a charged sphere (salt-free system).
