  We consider the simple exclusion process on Z x {0, 1}, that is, an
''horizontal ladder'' composed of 2 lanes, depending on 6 parameters. Particles
can jump according to a lane-dependent translation-invariant nearest neighbour
jump kernel, i.e. ''horizontally'' along each lane, and ''vertically'' along
the scales of the ladder. We prove that generically, the set of extremal
invariant measures consists of (i) translation-invariant product Bernoulli
measures; and, modulo translations along Z: (ii) at most two shock measures
(i.e. asymptotic to Bernoulli measures at $\pm$$\infty$) with asymptotic
densities 0 and 2; (iii) at most one (outside degenerate cases) shock measure
with a density jump of magnitude 1. We fully determine this set for a range of
parameter values. In fact, outside degenerate cases, there is at most one shock
measure of type (iii). Our results can be generalized in several directions
using the same approach and answer certain open questions formulated in
\cite{ligd} as a step towards the process on $\mathbb{Z}^2$.
