The mathematical theory of integrable Painlevé/Toda type systems sheds new light on the behaviour of solutions to the Poisson–Boltzmann equation for the potential due to a long rod-like macroion. We investigate here the case of symmetric electrolytes together with that of 1:2 and 2:1 salts. Small and large scale features are analysed, with particular emphasis on the low salinity regime. Analytical expansions are derived for several quantities relevant for polyelectrolyte theory, such as the Manning radius. In addition, accurate and practical expressions are worked out for the electrostatic potential, which improve upon previous work and cover the full range of radial distances.