  An inconsistency between the theoretical analysis and numerical calculations
of the relativistic $r$-modes puzzles the neutron star community since the
Kojima's finding of the continuous part in the $r$-mode oscillation spectrum in
1997. In this paper, after a brief review of the Newtonian $r$-mode theory and
of the literature devoted to the continuous spectrum of $r$-modes, we apply our
original approach to the study of relativistic oscillation equations. Working
within the Cowling approximation, we derive the general equations, governing
the dynamics of discrete relativistic $r$-modes for both barotropic
(isentropic) and nonbarotropic stars. A detailed analysis of the obtained
equations in the limit of extremely slow stellar rotation rate reveals that,
because of the effect of inertial reference frame-dragging, the relativistic
$r$-mode eigenfunctions and eigenfrequencies become {\it non-analytic}
functions of the stellar angular velocity, $\Omega$. We also derive the
explicit expressions for the $r$-mode eigenfunctions and eigenfrequencies for
very small values of $\Omega$. These expressions explain the asymptotic
behavior of the numerically calculated eigenfrequencies and eigenfunctions in
the limit $\Omega\to 0$. All the obtained $r$-mode eigenfrequencies take
discrete values in the frequency range, usually associated with the continuous
part of the spectrum. No indications of the continuous spectrum, at least in
the vicinity of the Newtonian $l=m=2$ $r$-mode frequency $\sigma=-4/3 \
\Omega$, are found.

High Energy Astrophysical Phenomena;General Relativity and Quantum Cosmology