  Rezzolla et al. [ApJ 531 (2000), L139; Phys. Rev. D 64 (2001), 104013; Phys.
Rev. D 64 (2001), 104014] draw attention to the second order secular drift
associated with r-modes and claimed that it should lead to magnetic field
enhancement and suppression of r-mode instability in magnetized neutron stars.
We critically revise these results. We present a particular second order r-mode
solution with vanishing secular drift, thus refuting a widely believed
statement that secular drift is an unavoidable feature of r-modes. This
non-drifting solution is not affected by magnetic field $B$, if $B\ll
B_{\mathrm{crit}}\approx 10^{17}\,(\nu/600\,\mathrm{Hz})$ G ($\nu$ is a spin
frequency) and does not lead to secular evolution of magnetic field. For
general second order r-mode solution the drift does not necessarily vanish, but
the solution can be presented as a superposition of two solutions: one
describes evolution of differential rotation in nonoscillating star (which
describes secular drift; for nonmagnetized star it is arbitrary stationary
rotation stratified on cylinders; for magnetized star differential rotation
evolves on the Alfv\'{e}n timescale and may lead to magnetic energy
enhancement), and another one is non-drifting r-mode solution mentioned above.
This representation allows us to conclude that enhancement of magnetic field
energy is limited by initial energy of differential rotation, which is much
less (for a factor $\propto \alpha^2$, where $\alpha$ is mode amplitude) than
the total energy of r-mode. Hence, magnetic field enhancement by drift cannot
suppress r-mode instability. Results can be generalized for any oscillation
mode in any medium, if this mode has non-drifting solution for $B=0$.
