  We consider the equivalence problem of four-dimensional semi-Riemannian
metrics with the $2$-dimensional Abelian Killing algebra. In the generic case
we determine a semi-invariant frame and a fundamental set of first-order scalar
differential invariants suitable for solution of the equivalence problem.
Genericity means that the Killing leaves are not null, the metric is not
orthogonally transitive (i.e., the distribution orthogonal to the Killing
leaves is non-integrable), and two explicitly constructed scalar invariants
$C_\rho$ and $\ell_{\mathcal C}$ are nonzero. All the invariants are designed
to have tractable coordinate expressions. Assuming the existence of two
functionally independent invariants, we solve the equivalence problem in two
ways. As an example, we invariantly characterise the Van den Bergh metric. To
understand the non-generic cases, we also find all $\Lambda$-vacuum metrics
that are generic in the above sense, except that either $C_\rho$ or
$\ell_{\mathcal C}$ is zero. In this way we extend the Kundu class to
$\Lambda$-vacuum metrics. The results of the paper can be exploited for
invariant characterisation of classes of metrics and for extension of the set
of known solutions of the Einstein equations.
