  It is shown that order-invariance of two-variable first-logic is decidable in
the finite. This is an immediate consequence of a decision procedure obtained
for the finite satisfiability problem for existential second-order logic with
two first-order variables ($\mathrm{ESO}^2$) on structures with two linear
orders and one induced successor. We also show that finite satisfiability is
decidable on structures with two successors and one induced linear order. In
both cases, so far only decidability for monadic $\mathrm{ESO}^2$ has been
known. In addition, the finite satisfiability problem for $\mathrm{ESO}^2$ on
structures with one linear order and its induced successor relation is shown to
be decidable in non-deterministic exponential time.
