  In this paper, we study the following anisotropic nonlinear Schr\"odinger
equation on the plane,
  \[ \begin{cases}
  {\rm i}\partial_t \Phi+\partial_{xx} \Phi -D_y^{2s} \Phi
+|\Phi|^{p-2}\Phi=0,&\quad (t,x,y)\in\mathbb{R} \times \mathbb{R}^2,
  \Phi(x,y,0)=\Phi_0(x,y),&\quad (x,y)\in\mathbb{R}^2,
  \end{cases}
  \]
  where $D_y^{2s}=\left(-\partial_{yy}\right)^s$ denotes the fractional
Laplacian with $0<s<1$ and $2<p<\frac{2(1+s)}{1-s}$. We first study the
existence of normalized solutions to this equation in the subcritical,
critical, and supercritical cases. To this aim, regularity results and a
Pohozaev type identity are necessary. Then, we determine the conditions under
which the solutions blow up. Furthermore, we demonstrate the existence of
boosted traveling waves when $s\geq1/2$ and their decay at infinity.
Additionally, for the delicate case $s=1/2$, we provide a non-existence result
of boosted traveling waves and we establish that there is no scattering for
small data. Finally, we also study normalized boosted travelling waves in the
mass subcritical case. Due to the nature of the equation, we do not impose any
radial symmetry on the initial data or on the solutions.
