  We discuss a generalized Schr\"odinger operator in $L^2(\mathbb{R}^d),
d=2,3$, with an attractive singular interaction supported by a
$(d-1)$-dimensional hyperplane and a finite family of points. It can be
regarded as a model of a leaky quantum wire and a family of quantum dots if
$d=2$, or surface waves in presence of a finite number of impurities if $d=3$.
We analyze the discrete spectrum, and furthermore, we show that the resonance
problem in this setting can be explicitly solved; by Birman-Schwinger method it
is cast into a form similar to the Friedrichs model.
