we consider the $S_{x}S_y$ case. Following relations among the coefficients must be satisfied

$$S_{x}S_y:
  \begin{cases}
c_1 = c_7\\
c_2 = c_6 = c_8 = c_{12}\\
c_3 = c_5 = c_9 = c_{11}\\
c_4 = c_{10}.
  \end{cases}
  \label{eq:11}$$

Then the task is to solve just four equations instead of twelve.

If we take , the equations have the form

$$S_{x}S_y:
  \begin{cases}
1. (\varepsilon+1)c_1 + 2c_2 & = 0\\
2. \quad c_1 + (\varepsilon+1)c_2 + c_3 & = 0\\
3. \qquad c_2 + (\varepsilon+1)c_3 + c_4 & = 0\\
4. \qquad \qquad 2c_3 + (\varepsilon+1) c_4 & = 0.
  \end{cases}
  \label{eq:12}$$

The determinant of has to be zero and therefore we obtain

$$S_x S_y: \varepsilon = -3, -2,0, 1.$$

In other symmetry cases we have

$$S_{x}A_y:
  \begin{cases}
c_1 = -c_7,\\
c_2 = c_6 =-c_8= -c_{12},\\
c_3 = c_5 =-c_9= -c_{11},\\
c_4=-c_{10}.
  \end{cases}
  \label{eq:14}$$

$$A_{x}S_y:
  \begin{cases}
c_1 = c_7 =0,\\
c_2 = - c_6 =c_8= -c_{12},\\
c_3 = -c_5 =c_9= -c_{11},\\
c_4=c_{10}=0.
  \end{cases}
  \label{eq:15}$$

$$A_{x}A_y:
  \begin{cases}
c_1 = c_7 =0,\\
c_2 = - c_6 =-c_8=c_{12},\\
c_3 = -c_5 =-c_9= c_{11},\\
c_4=c_{10}=0.
  \end{cases}
  \label{eq:16}$$

In the latter two cases there is $c_1=c_4=c_7=c_{10}=0$ due to the antisymmetry to axis $x$ ($A_x$). Therefore, only two equations 