\section{Parameter reconstruction for simple classical potentials}\label{aprec}
We provide here the results of the inversion procedure described in section \ref{secLinv} for a few classical potentials. Note 
that, for completeness, one should also replace the parameters within Eq.(\ref{mincondgen}) to fully determine the potential.

{\bf $\mathbb{Z}_3$-invariant potential ${\cal V}^S_{\mathbb{Z}_3}$:}\newline
The quartic doublet couplings are entirely determined by the Higgs mass-matrices:
\begin{equation}
 \label{lambdoub}\begin{cases}
 \lambda_1=\frac{1}{2v^2}\left[\frac{m^2_{h_i^0}S_{i1}^2}{\cos^2\beta}-
m^2_{a_i^0}P'^2_{i1}\tan^2\beta\right]\\
 \lambda_2=\frac{1}{2v^2}\left[\frac{m^2_{h_i^0}S_{i2}^2}{\sin^2\beta}-\frac{
m^2_{a_i^0}P'^2_{i1}}{\tan^2\beta}\right]\\
 \lambda_3=\frac{1}{2v^2}\left[2m_{H^{\pm}}^2+\frac{2m^2_{h_i^0}S_{i1}S_{i2}}{
\sin2\beta}- m^2_{a_i^0}P'^2_{i1}\right]\\
 \lambda_4=\frac{1}{v^2}\left[m^2_{a_i^0}P'^2_{i1}-m_{H^{\pm}}^2\right]
 \end{cases}
\end{equation}\noi
One degree of freedom remains, which can be chosen conveniently as the singlet v.e.v.\ $s$. One then obtains for the remaining 
parameters:
\begin{equation}
 \begin{cases}
 A_{ud}=\frac{1}{3}\left[\frac{\sin2\beta}{s}m^2_{a_i^0}P'^2_{i1}+\frac{1}{v}
m^2_{a_i^0}P'_{i1}P'_{i2}\right]\\
 \lambda_P^M=\frac{1}{3s}\left[\frac{\sin2\beta}{2s}m^2_{a_i^0}P'^2_{i1}-\frac{1
}{v}m^2_{a_i^0}P'_{i1}P'_{i2}\right]\\
 A_S=\frac{1}{3s}\left[\frac{v^2\sin^22\beta}{2s^2}m^2_{a_i^0}P'^2_{i1}-m^2_{
a_i^0}P'^2_{i2}-\frac{v\sin2\beta}{2s}m^2_{a_i^0}P'_{i1}P'_{i2}\right]\\
 \kappa^2=\frac{1}{4s^2}\left[m_{h_i^0}^2S_{i3}^2+\frac{1}{3}m^2_{a_i^0}P'^2_{i2
}-\frac{v^2\sin^22\beta}{3s^2}m^2_{a_i^0}P'^2_{i1}\right]\\
 \lambda_P^u=\frac{m_{h_i^0}^2S_{i2}S_{i3}}{2sv\sin\beta}+\frac{1}{3s\tan\beta}
\left[\frac{\sin2\beta}{s}m^2_{a_i^0}P'^2_{i1}-\frac{1}{2v}m^2_{a_i^0}P'_{i1}P'_
{i2}\right]\\
 \lambda_P^d=\frac{m_{h_i^0}^2S_{i1}S_{i3}}{2sv\cos\beta}+\frac{\tan\beta}{3s}
\left[\frac{\sin2\beta}{s}m^2_{a_i^0}P'^2_{i1}-\frac{1}{2v}m^2_{a_i^0}P'_{i1}P'_
{i2}\right]
 \end{cases}
\label{lambnew}\end{equation}\noi
In replacement of $s$, one may use any combination of these latter equations to define a new parameter. For instance,
\begin{multline}
 \varepsilon_{(a,b)}\equiv\frac{\lambda_{P}^M(a+b)}{a\lambda_P^u+b\lambda_P^d}\hspace{1cm}\Leftrightarrow\\
s=\frac{v\sin2\beta\,m^2_{a_i^0}P'^2_{i1}\left[1-2\varepsilon_{(a,b)}\left(\frac
{a}{\tan\beta}+b\tan\beta\right)\right]}{2m^2_{a_i^0}P'_{i1}P'_{i2}\left[1-\frac
{\varepsilon_{(a,b)}}{2}\left(\frac{a}{\tan\beta}+b\tan\beta\right)\right]
+3\varepsilon_{(a,b)}m_{h_i^0}^2S_{i3}\left(a\frac{S_{i1}}{\sin\beta}+b\frac{S_{
i2}}{\cos\beta}\right)}
\end{multline}\noi
$\varepsilon_{(a,b)}$ coincides with $\kappa/\lambda$ in the NMSSM at tree-level and may be regarded as a measurement of the 
breakdown of the Peccei-Quinn symmetry. Alternatively,
\begin{multline}
 \rho_{ud}\equiv\frac{\lambda_P^u}{\lambda_P^d}-1\hspace{1cm}\Leftrightarrow\\
s=\frac{2v\sin2\beta\,m_{a_i^0}^2P'^2_{i1}\left[1-(1+\rho_{ud})\tan^2\beta\right
]}{m_{a_i^0}^2P'_{i1}P'_{i2}\left[1-(1+\rho_{ud})\tan^2\beta\right]-3m_{h_i^0}
^2S_{i3}\left[\frac{S_{i1}}{\cos\beta}-(1+\rho_{ud})\tan^2\beta\frac{S_{i2}}{
\sin\beta}\right]}
\end{multline}
$\rho_{ud}$ vanishes at tree-level in the NMSSM and may represent another possibility.

{\bf Peccei-Quinn-invariant potential ${\cal V}^S_{PQ}$:}\newline
The system is fully determined by the mass matrices:
\begin{equation}
 \label{invPQ}\begin{cases}
 \lambda_1=\frac{1}{2v^2}\left[\frac{m^2_{h_i^0}S_{i1}^2}{\cos^2\beta}-
m^2_{a_2^0}P'^2_{21}\tan^2\beta\right]\\
 \lambda_2=\frac{1}{2v^2}\left[\frac{m^2_{h_i^0}S_{i2}^2}{\sin^2\beta}-\frac{
m^2_{a_2^0}P'^2_{21}}{\tan^2\beta}\right]\\
 \lambda_3=\frac{1}{2v^2}\left[2m_{H^{\pm}}^2+\frac{2m^2_{h_i^0}S_{i1}S_{i2}}{
\sin2\beta}- m^2_{a_2^0}P'^2_{21}\right]\\
 \lambda_4=\frac{1}{v^2}\left[m^2_{a_2^0}P'^2_{21}-m_{H^{\pm}}^2\right]\\
 A_{ud}=\frac{m^2_{a_2^0}P'_{21}P'_{22}}{v}\\
 s=\frac{v}{2}\sin{2\beta}\frac{P'_{21}}{P'_{22}}\\
 \kappa^2=\frac{(P'_{22}/P'_{21})^2}{v^2\sin^22\beta}\left[m_{h_i^0}^2S_{i3}
^2-m^2_{a_2^0}P'^2_{22}\right]\\
 \lambda_P^u=\frac{1}{2v^2\sin^2\beta}\frac{P'_{22}}{P'_{21}}\left[\frac{m_{
h_i^0}^2S_{i2}S_{i3}}{\cos{\beta}}+m^2_{a_2^0}P'_{21}P'_{22}\right]\\
 \lambda_P^d=\frac{1}{2v^2\cos^2\beta}\frac{P'_{22}}{P'_{21}}\left[\frac{m_{
h_i^0}^2S_{i1}S_{i3}}{\sin{\beta}}+m^2_{a_2^0}P'_{21}P'_{22}\right]
 \end{cases}
\end{equation}\noi

{\bf Peccei-Quinn'-invariant potential ${\cal V}^S_{PQ'}$:}\newline
The system is fully determined by the mass matrices:
\begin{equation}
 \label{invPQp}\begin{cases}
 \lambda_1=\frac{1}{2v^2}\left[\frac{m^2_{h_i^0}S_{i1}^2}{\cos^2\beta}-
m^2_{a_2^0}P'^2_{21}\tan^2\beta\right]\\
 \lambda_2=\frac{1}{2v^2}\left[\frac{m^2_{h_i^0}S_{i2}^2}{\sin^2\beta}-\frac{
m^2_{a_2^0}P'^2_{21}}{\tan^2\beta}\right]\\
 \lambda_3=\frac{1}{2v^2}\left[2m_{H^{\pm}}^2+\frac{2m^2_{h_i^0}S_{i1}S_{i2}}{
\sin2\beta}- m^2_{a_2^0}P'^2_{21}\right]\\
 \lambda_4=\frac{1}{v^2}\left[m^2_{a_2^0}P'^2_{21}-m_{H^{\pm}}^2\right]\\
 \lambda_P^M=\frac{m^2_{a_2^0}P'^2_{22}}{2v^2\sin{2\beta}}\\
 s=-v\sin{2\beta}\frac{P'_{21}}{P'_{22}}\\
 \kappa^2=\frac{m_{h_i^0}^2S_{i3}^2}{4v^2\sin^22\beta}\left(\frac{P'_{22}}{P'_{
21 }}\right)^2\\
 \lambda_P^u=\frac{1}{4v^2\sin^2\beta}\frac{P'_{22}}{P'_{21}}\left[m^2_{a_2^0}
P'_{21}P'_{22}-\frac{m_{h_i^0}^2S_{i2}S_{i3}}{\cos{\beta}}\right]\\
 \lambda_P^d=\frac{1}{4v^2\cos^2\beta}\frac{P'_{22}}{P'_{21}}\left[m^2_{a_2^0}
P'_{21}P'_{22}-\frac{m_{h_i^0}^2S_{i2}S_{i3}}{\sin{\beta}}\right]
 \end{cases}
\end{equation}\noi

{\bf nMSSM-inspired potential ${\cal V}^S_{T}$:}\newline
Although only twelve parameters are to be determined within the potential, application of the constraints
of Eq.(\ref{gencond}) leave one degree of freedom, due to the degenerescence of the CP-even and CP-odd 
singlet in this model: $m_{h_i^0}^2S_{i3}^2=m_{a_i^0}^2P'^2_{i2}$. We again choose $s$ to be this degree of 
freedom. The remaining parameters read:
\begin{equation}
 \label{invT}\begin{cases}
 \lambda_1=\frac{1}{2v^2}\left[\frac{m^2_{h_i^0}S_{i1}^2}{\cos^2\beta}-
m^2_{a_i^0}P'^2_{i1}\tan^2\beta\right]\\
 \lambda_2=\frac{1}{2v^2}\left[\frac{m^2_{h_i^0}S_{i2}^2}{\sin^2\beta}-\frac{
m^2_{a_i^0}P'^2_{i1}}{\tan^2\beta}\right]\\
 \lambda_3=\frac{1}{2v^2}\left[2m_{H^{\pm}}^2+\frac{2m^2_{h_i^0}S_{i1}S_{i2}}{
\sin2\beta}- m^2_{a_i^0}P'^2_{i1}\right]\\
 \lambda_4=\frac{1}{v^2}\left[m^2_{a_i^0}P'^2_{i1}-m_{H^{\pm}}^2\right]\\
 \lambda_T=\frac{v}{2}\sin{2\beta}m^2_{a_i^0}P'_{i1}P'_{i2}-sm^2_{a_i^0}P'^2_{i2
}\\
 m_{12}^2=m^2_{a_i^0}P'_{i1}P'_{i2}\frac{s}{v}-m^2_{a_i^0}P'^2_{i1}\frac{\sin{2\beta}}{2}\\
 A_{ud}=\frac{m^2_{a_i^0}P'_{i1}P'_{i2}}{v}\\
 \lambda_P^u=\frac{1}{2vs}\left[\frac{m_{h_i^0}^2S_{i2}S_{i3}}{\sin{\beta}}
+\frac{m^2_{a_i^0}P'_{i1}P'_{i2}}{\tan\beta}\right]\\
 \lambda_P^d=\frac{1}{2vs}\left[\frac{m_{h_i^0}^2S_{i1}S_{i3}}{\cos{\beta}}+m^2_
{a_i^0}P'_{i1}P'_{i2}\tan\beta\right]
 \end{cases}
\end{equation}\noi

