{\varepsilon}a_0 A$ respectively. Then, the 2-D Dysthe equation for $A$ is given by [@Trulsen1996]: $$\begin{gathered}
\label{eq:Dysthe}
  {\mathrm{i}}A_t = \frac{1}{8}A_{xx} - \frac{1}{4}A_{yy} + \frac{1}{2}|A|^2A \\ - {\mathrm{i}}{\varepsilon}\bigl(\frac{1}{16}A_{xxx} - \frac{3}{8}A_{xyy} - \frac{3}{2}|A|^2A_x + \frac{1}{4}A^2A^{\star}_x - {\mathrm{i}}A{\mathcal{H}}[|A|^2]\bigr),\end{gathered}$$ where ${\mathcal{H}}$ is the Hilbert transform and the subscripts $A_t = \partial_t A$, $A_x = \partial_x A$ denote partial derivatives with respect to $x$, $y$ and $t$ respectively, and $A^{\star}$ denotes complex conjugation. The envelope of the free surface $\eta$ relates to $A$ by a simple transformation that involves only $A$ and its derivatives (see [@Dysthe1979]). To ${\mathcal{O}}(1)$ in ${\varepsilon}$, the 2-D Dysthe equation reduces to the equation: $$\label{eq:hNLS}
  {\mathrm{i}}A_t = \frac{1}{8}A_{xx} - \frac{1}{4}A_{yy} + \frac{1}{2}|A|^2A.$$ By rescaling $A \to 2A$, $x \to 2\sqrt{2} x$ and the transverse coordinate $y \to 2y$, takes the form [@Sulem1999; @Pelinovsky2003]: $$\label{eq:adim}
  {\mathrm{i}}A_t = A_{xx} - A_{yy} + 2A|A|^2.$$ This equation admits the three invariants ${\mathbb{H}}$, ${\mathbb{A}}$ and ${\mathbb{M}}$, which have the meaning of energy, wave action and momentum respectively: $$\begin{aligned}
  {\mathbb{H}}&=& \iint_{{\mathds{R}}^2}\Bigl\{|A_y|^2 - |A_x|^2 + |A|^4\Bigr\}\,{\mathrm{d}}x\,{\mathrm{d}}y, \\
  {\mathbb{A}}&=& \iint_{{\mathds{R}}^2}|A|^2\,{\mathrm{d}}x\,{\mathrm{d}}y, \\
  {\mathbb{M}}&=& \frac{{\mathrm{i}}}{2}\iint_{{\mathds{R}}^2}\Bigl\{A{\boldsymbol{\nabla}}A^\star + A^\star{\boldsymbol{\nabla}}A\Bigr\}\,{\mathrm{d}}x\,{\mathrm{d}}y.\end{aligned}$$ Note that the total energy ${\mathbb{H}}$ is also the Hamiltonian for the equation. In the following, we will solve for a special class of standing 