In this paper, we study the following anisotropic nonlinear Schrödinger equation on the plane, \({\left\{ \begin{array}{ll} \textrm{i}\partial _t \Phi +\partial _{xx} \Phi -D_y^{2s} \Phi +|\Phi |^{p-2}\Phi =0,& \quad (t,x,y)\in \mathbb {R} \times \mathbb {R}^2,\\ \Phi (x,y,0)=\Phi _0(x,y),& \quad (x,y)\in {\mathbb {R}^2}, \end{array}\right. }\) 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 \(s\ge 1/2\) 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.
