Maxwell's equations as 2nd order DEQ

Maxwell-electrodynamics can be compactly written in terms of the 4-potential as \(\Box A_\mu = 4\pi/c\:\jmath_\mu\). What is missing in addition to this field equation to describe electrodynamics completely?

The field equation \(\Box A_\mu = 4\pi/c\:\jmath_\mu\) already assumes a gauge, Lorenz-gauge \(\partial^\mu A_\mu = 0\) to be exact.