\section{Trilinear Higgs-to-Higgs couplings}\label{hhhcoup}
In this appendix we give the physical trilinear Higgs-to-Higgs couplings $h_i^0
H^+ H^-$, $h_i^0 a_j^0 a_k^0$ and $h_i^0 h_j^0 h_k^0$ in the $\l$-representation
obtained from the general potential Eq.(\ref{genradpot}) and in the mass
representation from the restricted $\mathbb{Z}_3$ potential Eq.(\ref{Z3pot})
only (as in the general potential the results are cumbersome). In the following
the matrix $P_{ij}$ is defined as the $3\times3$ diagonalization matrix which
rotates the gauge eigenstates $(a_d^0, a_u^0, a_s^0)$ directly to the physical
basis $(a_1^0, a_2^0, G^0)$ such that,
\par
\begin{equation}
  \left(\begin{array}{c}a_1\\a_2\\G^0\end{array}\right) = 
\begin{bmatrix}
 P'_{11} \sinb & P'_{11} \cosb & P'_{12} \\
 P'_{21} \sinb & P'_{21} \cosb & P'_{22} \\
 \cosb & -\sinb & 0 
\end{bmatrix}
\left(\begin{array}{c}
a_d^0\\a_u^0\\a_S^0\end{array}\right) =
 \left[\begin{array}{ccc}
 P_{11} & P_{12} & P_{13}\\
 P_{21} & P_{22} & P_{23}\\
 P_{31} & P_{32} & P_{33}\\
\end{array}\right]\left(\begin{array}{c} a_d^0 \\ a_u^0 \\
a_S^0\end{array}\right)
\end{equation}\noi
with $P'$ defined from Eq.(\ref{condCPodd}). To cast the couplings in a more
compact form we also define the following mixing elements combinations,
\begin{eqnarray}
  (\Pi^A)^{a,b,c}_{i,j,k} &=& S_{ia}\left[P_{jb}P_{kc}+P_{jc}P_{kb} \right]\\
(\Pi^S)^{a,b,c}_{i,j,k}&=&S_{ia} S_{jb}S_{kc}+S_{ia}S_{jc}S_{kb}+S_{ib}
S_{ja}S_{kc}+S_{ib}S_{jc}S_{ka}+S_{ic} S_{ja}S_{kb}+S_{ic}S_{jb}S_{ka}
\end{eqnarray}\noi 

\subsection{Trilinear couplings in the \texorpdfstring{$\l$}{l}-representation}
\paragraph*{CP-even Higgs to charged Higgses \texorpdfstring{$h_i^0 H^+
H^-$}{hi0H+H-} coupling}
\begin{eqnarray}
\label{hHHlamb}
 g_{h_i^0 H^+ H^-} &=& \frac{\l_1 v \sinb \sbt S_{i1}}{\sqrt{2}}
+\frac{\l_2 v \cosb \sbt S_{i2}}{\sqrt{2}}
+\sqrt{2}\l_3 v [ \cosb^3 S_{i1} + \sinb^3 S_{i2}]
-\frac{(\l_4 + \l_5)v \sbt}{\sqrt{2}}\left [\sinb S_{i1}+\cosb S_{i2}
\right]\non \\
&-&\frac{\l_6 v \cosb}{\sqrt{2}}\left[\sbt S_{i1}-(1-3 c_{2\b})S_{i2} \right]
+\frac{\l_7 v \sinb}{\sqrt{2}}\left[(1+3 c_{2\b})S_{i1}-\sbt
S_{i2}\right]\non \\
&+&\frac{\sbt}{\sqrt{2}}\left[A_{ud}+{\tilde A}_{ud}
+4(\l^M_P+\lt^M_P+\l_M)s)\right]S_{i3}\non \\
&+&\sqrt{2}\left[\left(A_{ds}+(\l^d_P+2\lt^d_P)s\right)\sinb^2
+\left(A_{us}+(\l^u_P+2\lt^u_P)s\right)\cosb^2\right] S_{i3}
\end{eqnarray}\noi
\paragraph*{CP-even Higgs to 2 CP-odd Higgs \texorpdfstring{$h_i^0 a_i^0
a_j^0$}{hi0aj0ak0}}
\begin{eqnarray}
\label{haalamb}
 g_{h^0_i a_j^0 a_k^0} &=& \frac{\l_1 v
\cosb}{\sqrt{2}}(\Pi^A)^{1,1,1}_{i,j,k} 
+\frac{\l_2 v \sinb}{\sqrt{2}}(\Pi^A)^{2,2,2}_{i,j,k}
+\frac{(\l_3+\l_4)v}{\sqrt{2}}\left[\cosb(\Pi^A)^{1,2,2}_{i,j,k}+\sinb(\Pi^A)^{2
,1,1}_{ i,j,k}\right] \non \\
&-&\frac{\l_7 v \sinb}{\sqrt{2}}(\Pi^A)^{1,1,1}_{i,j,k}
-\frac{\l_6 v \cosb}{\sqrt{2}}(\Pi^A)^{2,2,2}_{i,j,k}
-\frac{v(\l_5 \cosb -\l_6\sinb)}{\sqrt{2}}(\Pi^A)^{1,2,2}_{i,j,k}
-\frac{v(\l_5 \sinb - \l_7\cosb)}{\sqrt{2}}(\Pi^A)^{2,1,1}_{i,j,k}\non \\
&-&\sqrt{2}v\left[(\l_5 \cosb+\l_6\sinb)(\Pi^A)^{2,1,2}_{i,j,k}
(\l_5 \sinb+\l_7\cosb)(\Pi^A)^{1,2,1}_{i,j,k}\right] \non \\
&+&\frac{A_{ud}-2\l^M_P s}{\sqrt{2}}(\Pi^A)^{1,2,3}_{i,j,k}
+\frac{A_{ud}-2\l^M_P s}{\sqrt{2}}(\Pi^A)^{2,1,3}_{i,j,k}
+\frac{A_{ud}+2\l^M_P s}{\sqrt{2}}(\Pi^A)^{3,1,2}_{i,j,k} \non \\
&-&\frac{\tilde A_{ud}-2\lt^M_P s}{\sqrt{2}}(\Pi^A)^{1,2,3}_{i,j,k}
-\frac{\tilde A_{ud}-2\lt^M_P s}{\sqrt{2}}(\Pi^A)^{2,1,3}_{i,j,k}
+\frac{\tilde A_{ud}+2(\lt^M_P+\l_M) s}{\sqrt{2}}(\Pi^A)^{3,1,2}_{i,j,k} \\
&+&\frac{A_{ds}+2\lt^d_P s}{\sqrt{2}}(\Pi^A)^{3,1,1}_{i,j,k}
+\frac{A_{us}+2\lt^u_P s}{\sqrt{2}}(\Pi^A)^{3,2,2}_{i,j,k}
-\frac{3A_S-\tilde A_S-3(2\k^2-3{\tilde\k}_S^2)
s}{3\sqrt{2}}(\Pi^A)^{3,3,3}_{i,j,k}\non \\
&-&\sqrt{2}(\l_P^M -\lt^M_P)v \left[\sinb(\Pi^A)^{3,1,3}_{i,j,k}
+\cosb(\Pi^A)^{3,2,3}_{i,j,k} \right]
+\frac{\l^d_P s}{\sqrt{2}}(\Pi^A)^{3,1,1}_{i,j,k}
+\frac{\l^u_P s}{\sqrt{2}}(\Pi^A)^{3,2,2}_{i,j,k}\non \\
&+&\frac{[(\lt^M_P-\l_M)\sinb-2\lt^d_P\cosb]v}{\sqrt{2}}(\Pi^A)^{1,3,3}_{i,j,k}
+\frac{[(\lt^M_P-\l_M)\cosb-2\lt^u_P\sinb]v}{\sqrt{2}}(\Pi^A)^{2,3,3}_{i,j,k}
\non 
\end{eqnarray}\noi
\paragraph*{Triple CP-even Higgs coupling \texorpdfstring{$h_i^0 h_j^0
h_k^0$}{hi0hj0hk0}}
\begin{eqnarray}
\label{hhhlamb}
 g_{h_i^0 h_j^0 h_k^0} &=&
\frac{\l_1 v\cosb}{\sqrt{2}}(\Pi^S)^{1,1,1}_{i,j,k}+
\frac{\l_2 v \sinb}{\sqrt{2}}(\Pi^S)^{2,2,2}_{i,j,k}+
\frac{(\l_3+\l_4)v}{\sqrt{2}}\left[\cosb(\Pi^S)^{1,2,2}_{i,j,k}+\sinb(\Pi^S)^{2,
1,1 }_{ i,j,k}\right] \non \\
&-&\frac{\l_6v}{\sqrt{2}}\left[\cosb(\Pi^S)^{2,2,2}_{i,j,k}+3\sinb(\Pi^S)^{1,2,2
} _{ijk}\right]-\frac{\l_7v}{\sqrt{2}}\left[\sinb(\Pi^S)^{1,1,1}_{i,j,k}
+3\cosb(\Pi^S)^{2,1,1}_{ijk}\right]\non \\
&+&\frac{(\l^d_P \cosb - \l^M_P\sinb)v}{\sqrt{2}}(\Pi^S)^{1,3,3}_{i,j,k}
+\frac{(\l^u_P \sinb  - \l^M_P\cosb)v}{\sqrt{2}}(\Pi^S)^{2,3,3}_{i,j,k}
+\frac{\l^d_P s}{\sqrt{2}}(\Pi^S)^{3,1,1}_{i,j,k}
+\frac{\l^u_P s}{\sqrt{2}}(\Pi^S)^{3,2,2}_{i,j,k}\non \\
&+&\frac{(2 {\tilde\l}^d_P\cosb-(\l_M+{\tilde\l}^M_P)\sinb)v}{\sqrt{2}}
(\Pi^S)^{1,3,3}_{i,j,k}
+\frac{(2 {\tilde\l}^u_P\sinb-(\l_M+{\tilde\l}^M_P)\cosb)v}{\sqrt{2}}
(\Pi^S)^{2,3,3}_{i,j,k} \\
&+&\sqrt{2}\left({\tilde\l}^d_P s(\Pi^S)^{3,1,1}_{i,j,k}+
{\tilde\l}^u_P s(\Pi^S)^{3,2,2}_{i,j,k}\right)
-\frac{A_{ud}+2\l^M_P s}{\sqrt{2}}(\Pi^S)^{3,1,2}_{i,j,k}
+\frac{A_S+6\k^2 s}{3\sqrt{2}}(\Pi^S)^{3,3,3}_{i,j,k}\non \\
&-&\frac{{\tilde A}_{ud}+2(\l_M+{\tilde\l}^M_P)
s}{\sqrt{2}}(\Pi^S)^{3,1,2}_{i,j,k}
+\frac{{\tilde A}_S+3(\k_S^2+{\tilde\k}_S^2)
s}{3\sqrt{2}}(\Pi^S)^{3,3,3}_{i,j,k}\non 
\end{eqnarray}\noi
\subsection{Trilinear couplings in the mass-representation for the
\texorpdfstring{$\mathbb{Z}_3$}{Z3}-conserving potential}
To obtain the mass representation we trade the $\mathbb{Z}_3$-conserving
$\l_i$'s of the couplings in the $\l$-representation (see previous
subsection) against the masses, mixing angles and
v.e.v.'s (Eq.(\ref{lambdoub},\ref{lambnew})) and set the remaining ones to zero.
\paragraph*{CP-even Higgs to charged Higgses
\texorpdfstring{$h_i^0 H^+
H^-$}{hi0H+H-} coupling}
\begin{eqnarray}
\label{hHHmass}
 g_{h_i^0 H^+ H^-} &=&
\frac{1}{v\sqrt{2}}\left\{m^2_{h^0_i}\left(\frac{\sb^2}{\cb}S_{i1}+\frac{
\cb^2 } { \sb }S_{i2}\right)+2 m_{H^\pm}^2(\cb S_{i1}+\sb S_{i2}) \non \right.\\
& &\left. -\frac{2 m_{a^0_j}^2 P'_{j1} P'_{j2}S_{i3}}{3 \sin 2
\b}-m_{a^0_j}^2P^{'\,2}_{j1}\left(\frac{S_{i1}}{\cb}+\frac{S_{i2}}{\sb}
-\frac{4}{3}\frac { v }{s}S_{i3}\right)\right\}
\end{eqnarray}\noi
\paragraph*{CP-even Higgs to 2 CP-odd Higgs \texorpdfstring{$h_i^0 a_i^0
a_j^0$}{hi0aj0ak0}}
\begin{eqnarray}
\label{haamass}
g_{h^0_i a_j^0 a_k^0} =\sum_{l=1}^3 \frac{m^2_{h^0_l}}{2\sqrt{2}} 
&\bigg \{&\frac{S_{l1}^2}{v\cosb}(\Pi^A)^{1,1,1}_{i,j,k}
+\frac{S_{l2}^2}{v\sinb}(\Pi^A)^{2,2,2}_{i,j,k}
+\frac{S_{l3}^2}{s}(\Pi^A)^{3,3,3}_{i,j,k}
+S_{l1}S_{l2}\bigg[
\frac{(\Pi^A)^{1,2,2}_{i,j,k}}{v\sinb}+\frac{(\Pi^A)^{2,1,1}_{i,j,k}}{v\cosb}
\bigg ]\non \\
&+ &
S_{l3}\bigg[S_{l1}\left(\frac{(\Pi^A)^{3,1,1}_{i,j,k}}{v\cosb}
+\frac{(\Pi^A)^{1,3,3}_{i,j,k}}{s}\right)
+S_{l2}\left(\frac{(\Pi^A)^{3,2,2} _{i,j,k}}{v\sinb}
+ \frac{(\Pi^A)^{2,3,3}_{i,j,k}}{s}\right) \bigg ] \non \\
+\sum_{l=1}^{2}\frac{m^2_{a^0_l}}{\sqrt{2}}&\bigg \{&
P_{l1}P_{l2}\bigg [
\frac{(\Pi^A)^{1,2,2}_{i,j,k}-\tb(\Pi^A)^{1,1,1}_{i,j,k}}{2v \cosb}
+\frac{(\Pi^A)^{2,1,1}_{i,j,k}-\tb^{-1}(\Pi^A)^{2,2,2}_{i,j,k}}{2v \sinb} \non\\
&+&\frac{4}{3 s} \bigg (
\tb(\Pi^A)^{3,1,1}_{i,j,k}
+\tb^{-1}(\Pi^A)^{3,2,2}_{i,j,k}
+(\Pi^A)^{3,1,2}_{i,j,k}\bigg ) \non \\
&+&\frac{v}{3s^2}\bigg(\sinb (\Pi^A)^{1,3,3}_{i,j,k}
+\cosb(\Pi^A)^{2,3,3}_{i,j,k}
-4(\sinb(\Pi^A)^{3,1,3}_{i,j,k}+\cosb(\Pi^A)^{3,2,3}_{i,j,k})
-\frac{2 v\sbt}{s}(\Pi^A)^{3,3,3}_{i,j,k} \bigg ]\non \\
&+&P_{l1}P_{l3}\bigg [\frac{1}{2v}\left(
\frac{(\Pi^A)^{1,2,3}_{i,j,k}}{\sinb}
+\frac{(\Pi^A)^{2,1,3}_{i,j,k}}{\sinb}
-\frac{(\Pi^A)^{3,1,2}_{i,j,k}}{3\sinb}
-\frac{(\Pi^A)^{3,1,1}_{i,j,k}}{3\cosb}\right) \\
&+&\frac{1}{3 s}\bigg ((\Pi^A)^{3,1,3}_{i,j,k}+\tb^{-1}(\Pi^A)^{3,2,3}_{i,j,k}
-(\Pi^A)^{1,3,3}_{i,j,k}
-\frac{\tb^{-1}(\Pi^A)^{2,3,3}_{i,j,k}}{2}
+\frac{v\cosb(\Pi^A)^{3,3,3}_{i,j,k}}{2s}\bigg )\bigg ]\non \\
&+&P_{l2}P_{l3}\bigg [\frac{1}{2v}\left(
\frac{(\Pi^A)^{1,2,3}_{i,j,k}}{\cosb}
+\frac{(\Pi^A)^{2,1,3}_{i,j,k}}{\cosb}
-\frac{(\Pi^A)^{3,1,2}_{i,j,k}}{3\cosb}
-\frac{(\Pi^A)^{3,2,2}_{i,j,k}}{3\sinb}\right)
\non \\
&+&\frac{1}{3 s}\bigg ((\Pi^A)^{3,2,3}_{i,j,k}+t_\b(\Pi^A)^{3,1,3}_{i,j,k}
-(\Pi^A)^{2,3,3}_{i,j,k}
-\frac{t_\b(\Pi^A)^{2,3,3}_{i,j,k}}{2}+\frac{v\sinb(\Pi^A)^{3,3,3}_{i,j,k}}{2s}
\bigg )\bigg ]\non \\
&+&\frac{P^{2}_{l3}}{2s}(\Pi^A)^{3,3,3}_{i,j,k}\bigg \}\non 
\end{eqnarray}\noi 
\paragraph*{Triple CP-even Higgs coupling \texorpdfstring{$h_i^0 h_j^0
h_k^0$}{hi0hj0hk0}}
\begin{eqnarray}
\label{hhhmass}
g_{h_i^0 h_j^0 h_k^0} &=& 
\sum_{l=1}^3 \frac{m^2_{h^0_l}}{2\sqrt{2}}\bigg\{\frac{S_{l1}^2
(\Pi^S)^{1,1,1}_{i,j,k}}{v
\cosb} +\frac{S_{l2}^2(\Pi^S)^{2,2,2}_{i,j,k}}{v\sinb}
+S_{l1}S_{l_2}\left[\frac{(\Pi^S)^{2,1,1}_{i,j,k}}{v\cosb}
+\frac{(\Pi^S)^{1,2,2}_{i,j,k}}{v\sinb}\right] \non \\
& &+S_{l1}S_{l_3}\left[\frac{(\Pi^S)^{3,1,1}_{i,j,k}}{v\cosb}
+\frac{(\Pi^S)^{1,3,3}_{i,j,k}}{s}\right]
+S_{l2}S_{l_3}\left[\frac{(\Pi^S)^{3,2,2}_{i,j,k}}{v\sinb}
+\frac{(\Pi^S)^{2,3,3}_{i,j,k}}{s}\right]
+\frac{S_{l3}^2(\Pi^S)^{3,3,3}_{i,j,k}}{s}\bigg\}\non \\
&+&\sum_{l=1}^2 \frac{m^2_{a^0_l}}{2\sqrt{2}}\bigg\{
P_{l1}^2 \bigg[
\frac{\tb^{-1}(\Pi^S)^{2,1,1}_{i,j,k}-(\Pi^S)^{1,1,1}_{i,j,k}}{v\cosb}\non \\
& &\qquad+\frac{4}{3s}\bigg(
(\Pi^S)^{3,1,1}_{i,j,k}
+\tb^{-1}(\Pi^S)^{3,1,2}_{i,j,k}
+\frac{v\cosb^2}{s}\bigg(\frac{(\Pi^S)^{1,3,3}_{i,j,k}}{2\sinb}
-\frac{v (\Pi^S)^{3,3,3}_{i,j,k}}{6s}\bigg)\bigg)\bigg] \\
&+&
P_{l2}^2 \bigg[
\frac{\tb(\Pi^S)^{1,2,2}_{i,j,k}-(\Pi^S)^{2,2,2}_{i,j,k}}{v\sinb}\non \\
& &\qquad+\frac{4}{3s}\bigg(
(\Pi^S)^{3,2,2}_{i,j,k}
+\tb(\Pi^S)^{3,1,2}_{i,j,k}
+\frac{v\sinb^2}{s}\bigg(\frac{(\Pi^S)^{2,3,3}_{i,j,k}}{2\cosb}
-\frac{v(\Pi^S)^{3,3,3}_{i,j,k}}{6s}\bigg)\bigg)\bigg]\non \\
&-&
\frac{P_{l1}P_{l3}}{3}
\bigg[\frac{(\Pi^S)^{1,1,1}_{i,j,k}}{v \cosb}
+\frac{(\Pi^S)^{3,1,2}_{i,j,k}}{v\sinb}
-\frac{\tb^{-1}(\Pi^S)^{1,3,3}_{i,j,k}}{s}
+\frac{v\cosb}{6s^2}(\Pi^S)^{3,3,3}_{i,j,k}\bigg]\non \\
&-&
\frac{P_{l2}P_{l3}}{3}
\bigg[\frac{(\Pi^S)^{2,2,2}_{i,j,k}}{v \sinb}
+\frac{(\Pi^S)^{3,1,2}_{i,j,k}}{v\cosb}
-\frac{\tb(\Pi^S)^{2,3,3}_{i,j,k}}{s}
+\frac{v\sinb}{6s^2}(\Pi^S)^{3,3,3}_{i,j,k}\bigg]+ 
\frac{P^{2}_{l3}(\Pi^S)^{3,3,3}_{i,j,k}}{9 s} \bigg\}\non
\end{eqnarray}\noi