  We present analytical solutions for the electrokinetics at a charged surface
with both non-zero Stern-layer conductance and finite chemical reaction rates.
We have recently studied the same system numerically [Werkhoven {\em et al.},
Phys. Rev. Lett. {\bf 120}, 264502 (2018)], and have shown that an applied
pressure drop across the surface leads to a non-trivial, laterally
heterogeneous surface charge distribution at steady state. In this work, we
linearise the governing electrokinetic equations to find closed expressions for
the surface charge profile and the generated streaming electric field. The main
results of our calculations are the identification of three important length
and time scales that govern the charge distribution, and consequently the
classification of electrokinetic systems into three distinct regimes. The three
governing time scales can be associated to (i) the chemical reaction, (ii)
diffusion in the Stern layer, and (iii) conduction in the Stern layer, where
the dominating (smallest) time scale characterises the regime. In the
reaction-dominated regime we find a constant surface charge with an edge
effect, and recover the Helmholtz-Smoluchowski equation. In the other two
regimes, we find that the surface charge heterogeneity extends over the entire
surface, either linearly (diffusion-dominated regime) or nonlinearly
(conduction-dominated regime).
