  The focussing anisotropic nonlinear Schr\"odinger equation \begin{align*}
\mathrm{i} u_t-\partial_{xx} u + (-\partial_{yy})^s u=|u|^{p-2}u \quad
\mbox{in}\ \mathbb{R} \times \mathbb{R}^2 \end{align*} is considered for
$0<s<1$ and $p>2$. Here the equation is of anisotropy, it means that dispersion
of solutions along $x$-axis and $y$-axis is different. We show that while
localized time-periodic waves, that are solutions in the form $u=e^{-\mathrm{i}
\omega t} \phi$, do not exist in the regime $p\geq p_s:=\frac{2(1+s)}{1-s}$,
they do exist in the complementary regime $2<p<p_s$. In fact, we construct them
variationally and we establish a number of key properties. Importantly, we
completely characterize their spectral stability properties. Our consideration
are easily extendable to the higher dimensional situation. We also show
uniqueness of these waves under a natural weak non-degeneracy assumption. This
assumption is actually removed for $s$ close to $1$, implying uniqueness for
the waves in the full range of parameters.
