  A previously-established model of steady one-dimensional two-ion
electrodiffusion across a liquid junction is reconsidered. It involves three
coupled first-order nonlinear ordinary differential equations, and has the
second-order Painlev\'e II equation at its core. Solutions are now grouped by
B\"acklund transformations into infinite sequences, partially labelled by two
B\"acklund invariants. Each sequence is characterized by evenly-spaced
quantized fluxes of the two ionic species, and hence evenly-spaced quantization
of the electric current-density. Finite subsequences of exact solutions are
identified, with positive ionic concentrations and quantized fluxes, starting
from a solution with zero electric field found by Planck, and suggesting an
interpretation as a ground state plus excited states of the system. Positivity
of ionic concentrations is established whenever Planck's charge-neutral
boundary-conditions apply. Exact solutions are obtained for the electric field
and ionic concentrations in well-stirred reservoirs outside each face of the
junction, enabling the formulation of more realistic boundary-conditions. In an
approximate form, these lead to radiation boundary conditions for Painlev\'e
II. Illustrative numerical solutions are presented, and the problem of
establishing compatibility of boundary conditions with the structure of
flux-quantizing sequences is discussed.
