![\[F:1\] (Color online) Indicators of the quadrupole collectivity (left) and of the octupole collectivity (right), as a function of the neutron number $N$ in the isotopic chain of Ra (circles) and Th (triangles): $(a)$ - Excitation energy of the first $2^+$ level; $(b)$ - Energy ratio $E(4^+)/E(2^+)$; $(c)$ - Excitation energy of the first level of the $K^\pi=0^-$ band, $J_0^\pi=1^-$ (open symbols) and of the lowest known level of other negative-parity bands, $J_0^\pi=2^-$ or $1^-_2$ (full symbols); $(d)$ - Energy ratio $E(1^-)/E(2^+)$. The horizontal line in the part $(b)$ shows the value (2.91) expected for the X(5) symmetry (from ref. [@bizzeti04]). ](biz_f1.ps){height="8.cm"}

The model for quadrupole – octupole vibrations
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\[S:3a\]Summary of the variable definitions
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The dynamical variables $a^{(\lambda )}_{\mu}$ ($\lambda=2,3;\ \mu
=-\lambda ... \lambda$), describing the quadrupole and octupole deformation in the intrinsic reference frame, are parametrized as $$\begin{aligned}
a^{(2)}_0 &=& \beta_2 \cos \gamma_2 \approx \beta_2 \nonumber\\*
a^{(2)}_1 &=& -  \frac{\sqrt{2}\ \beta_3}{\sqrt{\beta_2^2+2 
\beta_3^2}}\ v\ (\sin \varphi + i \cos \varphi )\cr 
a^{(2)}_2 &=& \sqrt{1/2}\ \beta_2\ \sin \gamma_2 - i 
\frac{\sqrt{5}\ \beta_3}{\sqrt{\beta_2^2+2 \beta_3^2}}\ u \sin \chi  \cr
a^{(3)}_0 &=& \beta_3 \ \cos \gamma_3 \approx \beta_3\cr
a^{(3)}_1 &=&\frac{\sqrt{5}\ \beta_2}{\sqrt{\beta_2^2+2 \beta_3^2}}\ v
\  (\sin \varphi + i \sin \varphi ) \cr
