**Abstract: ** We consider a quantum particle moving in the plane under an
influence of potential which can be modelled by delta interaction
supported by two parallel straight lines. We discuss spectral
properties of such a system with additional assumption that the
interaction asymptotically goes to a constant value. For example,
we formulate conditions guaranteeing either the existence of
discrete eigenvalues or Hardy-type inequalities. Furthermore, we
show that for a certain class of systems with a mirror symmetry
the embedded eigenvalues phenomena holds. Finally, we show that
the embedded eigenvalues turn into resonances after introducing a
small perturbation of the mirror symmetry.

The talk is based on a common work with David Krejcirik.