  The Brown-Ravenhall operator was initially proposed as an alternative to
describe the fermion-fermion interaction via Coulomb potential and subject to
relativity. This operator is defined in terms of the associated Dirac operator
and the projection onto the positive spectral subspace of the free Dirac
operator. In this paper, we propose to analyze a modified version of the
Brown-Ravenhall operator in two-dimensions. More specifically, we consider the
Brown-Ravenhall operator with a short-range attractive potential given by a
Bessel-Macdonald function (also known as $K_0$-potential) using the
Foldy-Wouthuysen unitary transformation. Initially, we prove that the
two-dimensional Brown-Ravenhall operator with $K_0$-potential is bounded from
below when the coupling constant is below a specified critical value (a
property also referred to as stability). A major feature of this model is the
fact that it does not cease to be bounded below even if the coupling constant
is above the specified critical value. We also investigate the nature of the
spectrum of this operator, in particular the location of the essential
spectrum, and the existence of eigenvalues, which are either isolated from the
essential spectrum or embedded in it.
