  The critical behavior of $d$-dimensional systems with $n$-component order
parameter $\bm{\phi}$ is studied at an $m$-axial Lifshitz point where a
wave-vector instability occurs in an $m$-dimensional subspace ${\mathbb R}^m$
($m{>}1$). Field theoretic renormalization group techniques are exploited to
examine the effects of terms in the Hamiltonian that break the rotational
symmetry of the Euclidean group ${\mathbb E}(m)$. The framework for considering
general operators of second order in $\bm{\phi}$ and fourth order in the
derivatives $\partial_\alpha $ with respect to the Cartesian coordinates
$x_\alpha $ of ${\mathbb R}^m$ is presented. For the specific case of systems
with cubic anisotropy, the effects of having an additional term,
$\sum_{\alpha=1}^m(\partial_\alpha^2\bm{\phi})^2$, are investigated in an
$\epsilon$ expansion about the upper critical dimension $d^{*}(m)=4+m/2$. Its
associated crossover exponent is computed to order $\epsilon^2$ and found to be
positive, so that it is a \emph{relevant} perturbation on a model isotropic in
${\mathbb R}^m$.
