this does not exclude the existence of nontrivial *nonlocalized* travelling wave patterns that may arise due to a balance between nonlinearities and wave dispersion in both directions under toric constraints.

An extended equation was derived by <span style="font-variant:small-caps;">Trulsen</span> & <span style="font-variant:small-caps;">Dysthe</span> (1996) [@Trulsen1996] and additionally to the classical hyperbolic (D’Alembert) operator, it contains also higher order dispersive and nonlinear terms. The generalization to the finite depth case leads to the Davey–Stewartson equations [@Benney1967; @Davey1974; @Ghidaglia1990]. Several important analytical solutions to the the Davey–Stewartson model were derived in [@Ebadi2011; @Ebadi2011a; @Yildirim2012; @Jafari2012].

For mathematical/numerical studies it is often convenient to restrict the attention to a particular class of solutions. For example, the very first mathematical description of plane permanent waves is known at least since G. <span style="font-variant:small-caps;">Stokes</span> (1847) [@Stokes1847]. Permanent waves can be periodic or localized in space. In this study we focus on bi-periodic time-harmonic solutions of the form $A = F(x - c_g t, y){\mathrm{e}}^{{\mathrm{i}}\omega t}$, where $A$ is the complex envelope and $F$ a complex function of its argument. These are stationary solutions in the frame of reference moving with the group speed $c_g$. In the physical domain, the associated wave surface displacements is that of standing 