## Appendix A  
To analyze the shape of the rogue wave solution, consider the normalized intensity function of \(w\) defined by \(F=\frac{|w|^{2}}{\rho^{2}}\). At the stationary point (0,0), the second derivatives are given by \(F_{tt}|_{(0,0)}=\frac{48a^{4}\left(b^{2}-a^{2}\right)}{\left(a^{2}+b^{2} \right)^{2}}\) and \(\left(F_{zz}F_{tt}-F_{zt}^{2}\right)|_{(0,0)}=\frac{256a^{10}\left(3a^{2}-b^ {2}\right)\left(a^{2}-3b^{2}\right)}{\left(a^{2}+b^{2}\right)^{4}}\). Similarly, for the evolution of \(u\) and \(v\), \(b\) is replaced by \(b-\delta\) and \(b+\delta\) respectively.  
From the real part and imaginary part of the dispersion relation Equation (3), note that  
\[\begin{array}{c}\left[2\left(\delta^{2}+3\sigma\rho^{2}\right)+3c\right]d^{ 2}=c^{3}+2\left(\delta^{2}+3\sigma\rho^{2}\right)c^{2}+\delta^{4}c+2\delta^{ 4}\sigma\rho^{2}\text{ and }\\ d^{2}=3c^{2}+4\left(\delta^{2}+3\sigma\rho^{2}\right)c+\delta^{4},\end{array} \tag{1}\]  
where \(c=a^{2}-b^{2}\) and \(d=2 ab\). It can be deduced that \(c\) is negative. Otherwise equating \(d^{2}\) in both equations will result in a cubic equation in \(c\), \(8c^{3}+16\left(\delta^{2}+3\sigma\rho^{2}\right)c^{2}+\left[2\delta^{4}+8 \left(\delta^{2}+3\sigma\rho^{2}\right)^{2}\right]c+2\delta^{6}+4\delta^{4} \sigma\rho^{2}=0\), which leads to a contradiction since the polynomial is positive if \(c\) is positive.  
Existence of EDRW with non-zero lower bound  
Without loss of generality, consider the case when both \(a\) and \(b\) are positive. If \(a=b\), then it is trivial from the second order derivative test that \(|v|\) attains minimum at (0,0). The case when \(a>b\) will lead to a contradiction, so it remains to consider the case when \(a<b\). From Equation (1), it can be shown that \(\delta^{2}-2c>\sqrt{c^{2}+d^{2}}\) which implies that \((b+\delta)^{2}-3a^{2}>2b\delta>0\). From the second order derivatives, it can be concluded that \(v\) is an EDRW when \(\Omega_{0}=a+ib\). On the other hand, \(u\) is an EDRW in the solution corresponding to the alternative root, \(\Omega_{0}=a-ib\). Hence, there must be a component with the configuration of an EDRW.  
Moreover, the amplitude of the dark rogue wave has a lower bound of about \(0.4q\). By considering the coefficient of the quadratic term in the dispersion relation, it can be shown that \(\delta>\sqrt{2}a\). With \(b>a\) and \((b+\delta)^{2}-3a^{2}>2b\delta\), it is easy to obtain the bound \(|v|>\frac{1}{1+\sqrt{2}}\rho\approx 0.4142\rho\). In particular, \(v\) cannot be a black rogue wave with zero minimum intensity.  
Non-existence of eye-shaped bright rogue wave  
Since \(a^{2}<b^{2}\), (0,0) is either a minimum point or a saddle point. Hence, \(w\) cannot be an eye-shaped bright rogue wave. Similarly, \(u\) and \(v\) cannot be an eye-shaped bright rogue wave. When \(a^{2}>(b+\delta)^{2}\), then \(a^{2}>b^{2}\) and from the above analysis \(3b^{2}>a^{2}\). It is then obvious that \(3(b+\delta)^{2}>a^{2}\) and from the sign of the Hessian, it can be concluded that (0,0) is a saddle point for \(|v|\) and \(v\) cannot be an eye-shaped bright rogue wave. Furthermore, by the symmetry between \(u\) and \(v\), it is trivial that \(u\) cannot possess such configuration either.