ne-forms $\{\text{d}\tau,\text{d}\varphi,\text{d}\psi\}$ via $$\mathbf{V}^{(m\,h\,k)}= V_{i}^{(m\,h\,k)}\text{d}x^i,\quad x\in\{\tau,\varphi,\psi\}\,.$$ The covector components are given by $$\begin{aligned}
  V_{j}^{(m\,h\,k)} \propto
  \begin{bmatrix}
    v^{(m\,h\,k)}_{\tau}(\sin \psi )^{-1} \\
    v^{(m\,h\,k)}_{\varphi} (\sin \psi )^{+0} \\
    v^{(m\,h\,k)}_{\psi} (\sin \psi )^{-1}
  \end{bmatrix}
  (\sin \psi )^{-h} e^{i [(h-k) \tau + m \varphi] +m \psi}
  \,,\end{aligned}$$ where $$\begin{aligned}
v_{\tau}^{(m\,h\,0)} &=   -\frac{1}{4 }\left(c_1 e^{- i \psi }+ 2c_1 e^{i \psi }-2 c_2 e^{ -i \psi } +4 c_3 e^{ i \psi }\right)\,, \\ {\nonumber}v_{\varphi}^{(m\,h\,0)} &= c_1 \,, \\ {\nonumber}v_{\psi}^{(m\,h\,0)} &= +\frac{1}{4 }\left(c_1 e^{- i \psi }+2 c_2 e^{ -i \psi } +4 c_3 e^{ i \psi }\right) \,,\end{aligned}$$ and $$\begin{aligned}
v_{\tau}^{(m\,h\,1)} =& -\frac{1}{4}\bigg\{
c_1 [2 (h+i m)e^{2 i \psi } +(3 h-i m-1)+(h-i m+1)e^{- 2i \psi }]-\\ {\nonumber}&-2c_2 [ (h+i m+1)+ (h-i m-1)e^{-2 i \psi }]
+4 c_3 [(h+i m-1)e^{2i \psi } +(h-i m+1)]
\bigg\}\,,\\  {\nonumber}v_{\varphi}^{(m\,h\,1)} =&  -2 c_1(m \sin \psi -h \cos \psi )\,, \\ {\nonumber}v_{\psi}^{(m\,h\,1)} =& +\frac{1}{4} \bigg\{
c_1 [(h+i m+1)+(h-i m-1)e^{-2 i \psi } ]+ 
2 c_2 [(h+i m+1)+(h-i m-1)e^{-2 i \psi } ]\\{\nonumber}&+4 c_3 [(h+i m-1)e^{2 i \psi} +(h-i m+1)]
\bigg\}\,.\end{aligned}$$

### Symmetric tensor bases {#app:tensor-basis-global}

The symmetric tensor bases in global coordinates can be decomposed using the dual basis one-forms $\{\text{d}\tau,\text{d}\varphi,\text{d}\psi\}$ via $$\mathbf{W}^{(m\,h\,k)} = W^{(m\,h\,k)}_{ij}\,\text{d}x^i\otimes \text{d}x^j,\quad x\in\{\tau,\varphi,\psi\}\,.$$ The tensor components are given by $$\begin{aligned}
  W_{ij}^{(m\,h\,k)} \propto
  \begin{bmatrix}
    w^{(m\,h\,k)}_{\tau\tau} (\sin \psi )^{-2} &
    w^{(m\,h\,k)}_{\tau\varphi} (\sin \psi )^{-1}&
    w^{(m\,h\,k)}_{\tau\psi} (\sin \psi )^{-2}\\
    * &
    w^{(m\,h\,k)}_{\varphi\varphi} (\sin \psi )^{+0}&
    w^{(m\,h\,k)}_{\varphi\psi} (\s