  The equations governing one-dimensional, steady-state electrodiffusion are
considered when there are arbitrarily many mobile ionic species present, in any
number of valence classes, possibly also with a uniform distribution of fixed
charges. Exact constant field solutions and new formulas of
Goldman-Hodgkin-Katz type are found. All of these formulas are exact, unlike
the usual approximate ones. Corresponding boundary conditions on the ionic
concentrations are identified. The question of uniqueness of constant field
solutions with such boundary conditions is considered, and is re-posed in terms
of an autonomous ordinary differential equation of order $n+1$ for the electric
field, where $n$ is the number of valence classes. When there are no fixed
charges, the equation can be integrated once to give the non-autonomous
equation of order $n$ considered previously in the literature including, in the
case $n=2$, the form of Painlev\'e's second equation considered first in the
context of electrodiffusion by one of us. When $n=1$, the new equation is a
form of Li\'enard's equation. Uniqueness of the constant field solution is
established in this case.
