## 5 Symmetry Protected BSCs  
In this section we present numerical solutions of Equation (3) for the symmetry protected BSCs with \(m=0,\beta=0\) embedded into the first diffraction continuum \(n=0\). They constitute the majority of the BSCs in the array. The symmetry protected BSCs are either pure TE spherical vector modes (Type I in Table 1) with \(a^{0}_{2k}\
eq 0,b^{0}_{k}=0\) or TM spherical vector modes (Type II in Table 1) with \(a^{0}_{k}=0,b^{0}_{2k}\
eq 0\). We show that the symmetry protected BSCs are symmetrically mismatched to the first open continuum. Below we present numerical solutions for Type I BSCs with accuracy of \(10^{-4}\):  
\[k_{0}=4.24,\ \ R=0.3,\epsilon=12,\ \ a^{0}_{l}=\left(\begin{array}{c}0\\ 0.7563-0.6542i\\ 0\end{array}\right),\ \ b^{0}_{l}=0,l\geq 1 \tag{24}\]  
and for Type II as  
\[k_{0}=4.7504,\ \ R=0.3,\epsilon=15,\ \ a^{0}_{l}=0,\ b^{0}_{l}=\left( \begin{array}{c}0\\ -0.6017+0.7988i\\ 0\\ 0.0004-0.0006i\end{array}\right),l\geq 1, \tag{25}\]  
Patterns of EM fields and EM force lines are shown in Figure 2. Hereinafter we plot only real parts of electromagnetic fields. Other patterns of the symmetry protected BSCs the reader can find in  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Type I** & **Type II** \\ \hline \(E_{x,y}(-z)=(-1)^{m+1}E_{x,y}(z)\) & \(E_{x,y}(-z)=(-1)^{m}E_{x,y}(z)\) \\ \hline \(E_{z}(-z)=(-1)^{m}E_{z}(z)\) & \(E_{z}(-z)=(-1)^{m+1}E_{z}(z)\) \\ \hline \(H_{x,y}(-z)=(-1)^{m}H_{x,y}(z)\) & \(H_{x,y}(-z)=(-1)^{m+1}H_{x,y}(z)\) \\ \hline \(H_{z}(-z)=(-1)^{m+1}H_{z}(z)\) & \(H_{z}(-z)=(-1)^{m}H_{z}(z)\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Symmetry properties of the eigenmodes with \(\beta=0\).  
Ref. [56]. One can see from Figure 2 that in the BSC of Type I (II) electric (magnetic) force lines are parallel the sphere surface.  
The symmetry protected Type I and Type II BSCs have qualitatively similar field structure with respect to \(\mathbf{E}\leftrightarrow\mathbf{H}\) but are not degenerate because of different boundary conditions for \(\mathbf{E}\) and \(\mathbf{H}\) at the sphere surface. From Table 2 one can see why the eigenmodes (24) and (25) are protected by symmetry against decay into the diffraction continuum \(m=0,n=0\). From Equations (18) and (19) we obtain that the TE/TM continuum with \(k_{z,0}=0\) (\(\beta=0\)) has the only \(H_{z}/E_{z}\
eq 0\) independent of \(z\). The Type I BSC has \(E_{z}=0\) and odd \(H_{z}\) so that these type of BSCs is symmetrically mismatched to both TE and TM continua. The Type II BSC has odd \(E_{z}\) and \(H_{z}=0\) to decouple from the both TE and TM continua.  
Besides the fully symmetry protected BSCs from the third row in Table 1 (\(a_{2k}^{0},0\)) and (\(0,b_{2k}^{0}\)), we found a partially symmetry protected Type II BSC (\(a_{2k+1}^{0},0\)) from the fourth row of Table 1:  
\[k_{0}=2.934,\ R=0.4805,\ \epsilon=15,\ \ a_{l}^{0}=\left(\begin{array}{c}0. 6826+0.0332i\\ 0\\ -0.7291-0.0354i\\ 0\\ -0.0008\end{array}\right),\ \ b_{l}^{0}=0,l\geq 1, \tag{26}\]  
however the Type I BSCs with \((0,b_{2k+1}^{0})\) were not revealed in our computations. The BSC (26) is symmetrically mismatched relative only to the TM continuum. Zero coupling of this BSC with the TE continuum can be achieved by tuning the radius of spheres. Patterns of EM fields and EM force lines are shown in Figure 3.  
Figure 2: Patterns of the symmetry protected TE BSC (transverse electric bound state in the continuum) (24) (**a**) and TM BSC (25) (**b**). Left panels show the real parts of electromagnetic (EM) field components, right panels show the electric force lines in red and magnetic force lines in blue.