Appendix A The Derivation of System (14)
The first three equations in system (14) can be easily acquired. We here just derive the fourth equation of (14) and the last equation can be derived symmetrically. Putting (10) and (11) into the fourth equation of (7) yields
$$\begin{aligned}&c\frac{\sin \sigma }{\sin \omega }{{\tilde{\partial }}}^+\bigg (c\frac{\cos \sigma }{\sin \omega }\bigg ) -c\frac{\cos \sigma }{\sin \omega }{{\tilde{\partial }}}^+\bigg (c\frac{\sin \sigma }{\sin \omega }\bigg ) -c\frac{\cos \omega }{\sin \omega }\frac{{{\tilde{\partial }}}^+p}{\gamma p} \nonumber \\& =\frac{c^2}{\sin ^2\omega }\sin \alpha \cot \theta +uv\sin \alpha -uw\cos \alpha . \end{aligned}$$
(A1)
Recalling the definition of B gives
$$\begin{aligned} B=\frac{q^2}{2}+\frac{c^2}{\gamma -1}&=\bigg (\frac{1}{2\sin ^2\omega }+\frac{1}{\gamma -1}\bigg )c^2 =\frac{\gamma (\kappa +\sin ^2\omega )}{2\kappa \sin ^2\omega }\cdot \frac{p}{\rho }. \end{aligned}$$
Then we use the entropy function \(S=p\rho ^{-\gamma }\) to find that
$$\begin{aligned} \ln p=\frac{\gamma }{2\kappa }\bigg (\ln B-\frac{1}{\gamma }\ln S-\ln \frac{\gamma (\kappa +\sin ^2\omega )}{2\kappa \sin ^2\omega }\bigg ). \end{aligned}$$
Thus we get
$$\begin{aligned} \frac{1}{\gamma p}{\tilde{\partial }}^+p=\frac{1}{2\kappa }{\tilde{\partial }}^+\bigg (\ln B-\frac{1}{\gamma }\ln S\bigg )+\frac{{\tilde{\partial }}^+\sin \omega }{(\kappa +\sin ^2\omega )\sin \omega }. \end{aligned}$$
Inserting the above into (A1) leads to
$$\begin{aligned}&c^2\frac{\sin \sigma }{\sin \omega }\cdot \frac{-\sin \sigma \sin \omega {\tilde{\partial }}^+\sigma -\cos \sigma {\tilde{\partial }}^+\sin \omega }{\sin ^2\omega } -c^2\frac{\cos \sigma }{\sin \omega }\cdot \frac{\cos \sigma \sin \omega {\tilde{\partial }}^+\sigma -\sin \sigma {\tilde{\partial }}^+\sin \omega }{\sin ^2\omega }\\& -c^2\cot \omega \cdot \bigg [\frac{1}{2\kappa }{\tilde{\partial }}^+\bigg (\ln B-\frac{1}{\gamma }\ln S\bigg )+\frac{{\tilde{\partial }}^+\sin \omega }{(\kappa +\sin ^2\omega )\sin \omega }\bigg ]\\&=\frac{c^2}{\sin ^2\omega }\sin \alpha \cot \theta +uc\frac{\cos \sigma }{\sin \omega }\sin \alpha -uc\frac{\sin \sigma }{\sin \omega }\cos \alpha , \end{aligned}$$
and doing a simplification arrives at
$$\begin{aligned}&{\tilde{\partial }}^+\theta +\frac{\cos \omega }{\kappa +\sin ^2\omega }{\tilde{\partial }}^+\sin \omega -\frac{\sin (2\omega )}{4\kappa }{\tilde{\partial }}^+\bigg (\frac{1}{\gamma }\ln S-\ln B\bigg ) \\& =-G\sqrt{\kappa +\sin ^2\omega }\sin \omega -\sin \alpha \cot \theta . \end{aligned}$$
Appendix B The Derivations of System (EQ-a)–(EQ-b)
The derivations are based on the commutator relations (21). We first provide some relations by simple calculations
$$\begin{aligned} {{\tilde{\partial }}}^+{{\tilde{\partial }}}^0G&=-\frac{2\sin ^2\omega }{\sqrt{\kappa +\sin ^2\omega }}(X+H)(1+\kappa G^2)+\frac{2\kappa GF}{\sqrt{\kappa +\sin ^2\omega }}, \\ \cos \omega {{\tilde{\partial }}}^0\alpha -{{\tilde{\partial }}}^+\sigma&=\sin \omega \cos \omega (X+Y)+\frac{\sin \omega (\kappa +\sin ^2\omega )}{\cos \omega }(X+Y)\\&\quad +2G\sqrt{\kappa +\sin ^2\omega }\sin \omega +\sin \omega \cos \sigma \cot \theta , \\ \cos \omega {{\tilde{\partial }}}^+\sigma -{{\tilde{\partial }}}^0\alpha&=-2\sin \omega \cos ^2\omega X -\sin \omega (Y-X)-\frac{\sin \omega (\kappa +\sin ^2\omega )}{\cos ^2\omega }(X+Y)\\&\quad -G\sqrt{\kappa +\sin ^2\omega }\sin \omega \cos \omega -\frac{G\sqrt{\kappa +\sin ^2\omega }\sin \omega }{\cos \omega } \\&\quad +\sin \sigma \cot \theta -\sin \alpha \cos \omega \cot \theta , \end{aligned}$$
and
$$\begin{aligned} {{\tilde{\partial }}}^0\bigg (\frac{1}{4\kappa \gamma }\ln S-\frac{1}{4\kappa }\ln B\bigg )&=\frac{G}{2\sqrt{\kappa +\sin ^2\omega }}, \\ {{\tilde{\partial }}}^+{{\tilde{\partial }}}^0\bigg (\frac{1}{4\kappa \gamma }\ln S-\frac{1}{4\kappa }\ln B\bigg )&=\frac{F-2G\sin ^2\omega (X+H)}{2\sqrt{\kappa +\sin ^2\omega }}. \end{aligned}$$
Making use of the commutator relation \(({{\tilde{\partial }}}^0, {{\tilde{\partial }}}^+)\) in (21), we achieve
$$\begin{aligned} {\tilde{\partial }}^0H&={{\tilde{\partial }}}^0{{\tilde{\partial }}}^+\bigg (\frac{1}{4\kappa \gamma }\ln S-\frac{1}{4\kappa }\ln B\bigg )\\&={{\tilde{\partial }}}^+{{\tilde{\partial }}}^0\bigg (\frac{1}{4\kappa \gamma }\ln S-\frac{1}{4\kappa }\ln B\bigg )+\frac{\cos \omega {{\tilde{\partial }}}^0\alpha -{{\tilde{\partial }}}^+\sigma }{\sin \omega }{{\tilde{\partial }}}^+\bigg (\frac{1}{4\kappa \gamma }\ln S-\frac{1}{4\kappa }\ln B\bigg )\\&\quad +\frac{\cos \omega {{\tilde{\partial }}}^+\sigma -{{\tilde{\partial }}}^0\alpha }{\sin \omega }{{\tilde{\partial }}}^0\bigg (\frac{1}{4\kappa \gamma }\ln S-\frac{1}{4\kappa }\ln B\bigg )\\&\quad -\cot \theta \bigg [\cos \sigma H-\cos \alpha \cdot {{\tilde{\partial }}}^0\bigg (\frac{1}{4\kappa \gamma }\ln S-\frac{1}{4\kappa }\ln B\bigg )\bigg ]. \end{aligned}$$
Putting the previous relations into the above and simplifying the resulting gets
$$\begin{aligned} {\tilde{\partial }}^0H =&\frac{F}{2\sqrt{\kappa +\sin ^2\omega }}-\frac{G(\kappa +1)(X+Y)}{2\cos ^2\omega \sqrt{\kappa +\sin ^2\omega }} +\frac{GH(2\kappa +\sin ^2\omega )}{\sqrt{\kappa +\sin ^2\omega }}\\&+\frac{(\kappa +1)(X+Y)H}{\cos \omega }-\frac{(1+\cos ^2\omega )G^2}{2\cos \omega }, \end{aligned}$$
which is the equation of H in (EQ-a). Similarly, one has the equation of F in (EQ-a)
$$\begin{aligned} {\tilde{\partial }}^0F=&{\tilde{\partial }}^+{\tilde{\partial }}^0G +\frac{\cos \omega {\tilde{\partial }}^0\alpha -{\tilde{\partial }}^+\sigma }{\sin \omega }F +\frac{\cos \omega {\tilde{\partial }}^+\sigma -{\tilde{\partial }}^0\alpha }{\sin \omega }{\tilde{\partial }}^0G -\cot \theta (\cos \sigma F-\cos \alpha {{\tilde{\partial }}}^0G)\\ =&-\frac{2\sin ^2\omega (1+\kappa G^2)H}{\sqrt{\kappa +\sin ^2\omega }}-\frac{(\kappa +1)(1+\kappa G^2)(X+Y)}{\cos ^2\omega \sqrt{\kappa +\sin ^2\omega }}+\frac{2GF(2\kappa +\sin ^2\omega )}{\sqrt{\kappa +\sin ^2\omega }}\\&+\frac{(\kappa +1)(X+Y)F}{\cos \omega }-\frac{(1+\cos ^2\omega )(1+\kappa G^2)G}{\cos \omega }. \end{aligned}$$
We now derive the equation of X. By the commutator relation \(({\tilde{\partial }}^-,{\tilde{\partial }}^+)\) in (21), it suggests that
$$\begin{aligned}&{\tilde{\partial }}^-{\tilde{\partial }}^+\sigma -{\tilde{\partial }}^+{\tilde{\partial }}^-\sigma \nonumber \\& =\frac{\cos (2\omega ){\tilde{\partial }}^-\alpha -{\tilde{\partial }}^+\beta }{\sin (2\omega )}{\tilde{\partial }}^+\sigma +\frac{\cos (2\omega ){\tilde{\partial }}^+\beta -{\tilde{\partial }}^-\alpha }{\sin (2\omega )}{\tilde{\partial }}^-\sigma -\cot \theta (\cos \beta {{\tilde{\partial }}}^+\sigma -\cos \alpha {{\tilde{\partial }}}^-\sigma ), \end{aligned}$$
(B1)
and
$$\begin{aligned}&{\tilde{\partial }}^-X-{\tilde{\partial }}^+Y \nonumber \\& =\frac{\cos (2\omega ){\tilde{\partial }}^-\alpha -{\tilde{\partial }}^+\beta }{\sin (2\omega )}X +\frac{\cos (2\omega ){\tilde{\partial }}^+\beta -{\tilde{\partial }}^-\alpha }{\sin (2\omega )}Y-\cot \theta (\cos \beta X-\cos \alpha Y). \end{aligned}$$
(B2)
Moreover, one differentiates (16) along the directions \({{\tilde{\partial }}}^-\) and \({{\tilde{\partial }}}^+\) to find that
$$\begin{aligned} {\tilde{\partial }}^-{\tilde{\partial }}^+\sigma =&-2\cos (2\omega )X{{\tilde{\partial }}}^-\omega -\sin (2\omega ){{\tilde{\partial }}}^-X-{{\tilde{\partial }}}^-G\sqrt{\kappa +\sin ^2\omega }\sin \omega -\frac{G\sin ^2\omega \cos \omega {{\tilde{\partial }}}^-\omega }{\sqrt{\kappa +\sin ^2\omega }}\\&-G\sqrt{\kappa +\sin ^2\omega }\cos \omega {{\tilde{\partial }}}^-\omega -\cos \alpha \cot \theta {{\tilde{\partial }}}^-\alpha +\sin \alpha \csc ^2\theta {{\tilde{\partial }}}^-\theta ,\\ {\tilde{\partial }}^+{\tilde{\partial }}^-\sigma =&2\cos (2\omega )Y{{\tilde{\partial }}}^+\omega +\sin (2\omega ){{\tilde{\partial }}}^+Y+{{\tilde{\partial }}}^+G\sqrt{\kappa +\sin ^2\omega }\sin \omega +\frac{G\sin ^2\omega \cos \omega {{\tilde{\partial }}}^+\omega }{\sqrt{\kappa +\sin ^2\omega }}\\&+G\sqrt{\kappa +\sin ^2\omega }\cos \omega {{\tilde{\partial }}}^+\omega -\cos \beta \cot \theta {{\tilde{\partial }}}^+\beta +\sin \beta \csc ^2\theta {{\tilde{\partial }}}^+\theta . \end{aligned}$$
We insert the above into (B1) and do a simplification to deduce
$$\begin{aligned}&\sin (2\omega )({{\tilde{\partial }}}^-X+{{\tilde{\partial }}}^+Y)\\& =-2\cos (2\omega )(X{{\tilde{\partial }}}^-\omega +Y{{\tilde{\partial }}}^+\omega ) -\sin (2\omega )(1+\kappa G^2)-\frac{G\cos \omega (\kappa +2\sin ^2\omega )}{\sqrt{\kappa +\sin ^2\omega }}({{\tilde{\partial }}}^+\omega +{{\tilde{\partial }}}^-\omega )\\& \quad-\cot \theta (\cos \alpha {{\tilde{\partial }}}^-\alpha -\cos \beta {{\tilde{\partial }}}^+\beta )+\csc ^2\theta (\sin \alpha {{\tilde{\partial }}}^-\theta -\sin \beta {{\tilde{\partial }}}^+\theta )\\&\quad +\cot \theta (\cos \beta {{\tilde{\partial }}}^+\sigma -\cos \alpha {{\tilde{\partial }}}^-\sigma ) -\frac{\cos (2\omega ){{\tilde{\partial }}}^-\alpha -{{\tilde{\partial }}}^+\beta }{\sin (2\omega )}{{\tilde{\partial }}}^+\sigma -\frac{\cos (2\omega ) {{\tilde{\partial }}}^+\beta -{{\tilde{\partial }}}^-\alpha }{\sin (2\omega )}{{\tilde{\partial }}}^-\sigma , \end{aligned}$$
from which, (B2) and (20), we acquire
$$\begin{aligned} {\tilde{\partial }}^-X =&\frac{\cos (2\omega ){{\tilde{\partial }}}^-\alpha -{{\tilde{\partial }}}^+\beta -\cos (2\omega ){{\tilde{\partial }}}^-\omega }{\sin (2\omega )}X -\frac{\cos (2\omega ){{\tilde{\partial }}}^+\omega }{\sin (2\omega )}Y\nonumber \\&-\frac{\cot \theta }{2}(\cos \beta X-\cos \alpha Y) -\frac{G\sqrt{\kappa +\sin ^2\omega }(\kappa +2\sin ^2\omega )}{2\cos \omega }(X+Y)-\frac{1+\kappa G^2}{2} \nonumber \\&-\frac{G^2(\kappa +2\sin ^2\omega )}{2}+\frac{\csc ^2\theta }{2} +\frac{\cot \theta }{2\sin (2\omega )}(\cos \beta {{\tilde{\partial }}}^+\sigma -\cos \alpha {{\tilde{\partial }}}^-\sigma ) \nonumber \\&+\frac{G\sqrt{\kappa +\sin ^2\omega }\sin \omega }{2\sin ^2(2\omega )}[(\cos (2\omega )+1)({{\tilde{\partial }}}^-\alpha -{{\tilde{\partial }}}^+\beta )]. \end{aligned}$$
(B3)
Furthermore, we apply (16) to calculate
$$\begin{aligned}&{{\tilde{\partial }}}^-\alpha -{{\tilde{\partial }}}^+\beta \\&=\frac{2\sin \omega (\kappa +1)}{\cos \omega }(X+Y)+4G\sqrt{\kappa +\sin ^2\omega } \sin \omega +2\cot \theta \cos \sigma \sin \omega ,\\&\cos \beta {{\tilde{\partial }}}^+\sigma -\cos \alpha {{\tilde{\partial }}}^-\sigma \\& =-\sin (2\omega )(\cos \beta X+\cos \alpha Y)-G\sqrt{\kappa +\sin ^2\omega }\cos \sigma \sin (2\omega ) -\cot \theta \sin (2\omega ),\\&\frac{\cos (2\omega ){{\tilde{\partial }}}^-\alpha -{{\tilde{\partial }}}^+\beta -\cos (2\omega ){{\tilde{\partial }}}^-\omega }{\sin (2\omega )} \\& =\cos (2\omega )Y+X +G\sqrt{\kappa +\sin ^2\omega }\cos \omega +\frac{\kappa +\sin ^2\omega }{\cos ^2\omega }(X+H) +\cot \theta \cos \beta .
\end{aligned}$$
It follows by combining with the above and (B3) that
$$\begin{aligned} {\tilde{\partial }}^-X=&\frac{\kappa +\sin ^2\omega }{\cos ^2\omega }(X+H)(X+Y)+X[\cos (2\omega )Y+X +G\sqrt{\kappa +\sin ^2\omega }\cos \omega ]\\&-2(\kappa +\sin ^2\omega )(X+H)Y+\frac{\cos (2\omega )\sqrt{\kappa +\sin ^2\omega }}{2\cos \omega }G(X+Y) -\frac{G^2}{2}+\frac{1}{2}, \end{aligned}$$
which is the equation of X in (EQ-b), and the equation of Y can be derived analogously.
Appendix C The Expressions of \(b_{ij}\) in System (45)
We here list the detailed expressions of \(b_{ij}(i=0,\cdots ,5;\, j=1,2,3)\) in system (45). The expressions of \(b_{01}\) and \(b_{11}\) are
$$\begin{aligned} b_{01}=&-\frac{1}{2\hbar ^2({\widetilde{X}}+{\widetilde{H}}+\Phi _0)}\big(2t\sqrt{1-t^2}\theta _{0}' {\widetilde{X}}+\hbar \theta _{0}'{\widetilde{G}}+\theta _{0}'\Phi _1+t\cos \zeta -\sqrt{1-t^2}\sin \zeta \big ),\\ b_{11}=&-\frac{1}{2\hbar ^2({\widetilde{X}}+{\widetilde{H}}+\Phi _0)}\bigg (2G_0't\sqrt{1-t^2}{\widetilde{X}} +G_0'\hbar {\widetilde{G}}+{\widetilde{F}}+F_0+\frac{(1+KG_{0}^2)t}{\sqrt{\kappa +1}}+G_0'\Phi _1\bigg ). \end{aligned}$$
The expressions of \(b_{2j}\ (j=1,2,3)\) are
$$\begin{aligned} b_{21}&=\frac{\kappa +1}{2\hbar ^2\sqrt{\kappa +1-t^2}},\quad b_{23}=-\frac{(\kappa +1)({\widetilde{H}}+H_0+\frac{G_0t}{2\sqrt{\kappa +1}})}{\hbar ^2},\\ b_{22}&=-\frac{1}{\hbar ^2}\big (b_{221}{\widetilde{X}}+b_{222}{\widetilde{Y}} +b_{223}{\widetilde{G}}+b_{224}{\widetilde{H}} +b_{225} +b_{226} \big ), \end{aligned}$$
where
$$\begin{aligned} b_{221}=&\sqrt{1-t^2}\bigg(H_0'+\frac{G_0't}{2\sqrt{\kappa +1}}\bigg) -\frac{G_0\hbar ^2}{2\sqrt{\kappa +1}\sqrt{\kappa +1-t^2} (\sqrt{\kappa +1}+\sqrt{\kappa +1-t^2})}\\&-\frac{G_0(\kappa +2-t^2)}{2\sqrt{\kappa +1-t^2}}, \\ b_{222}=&-\sqrt{1-t^2}\bigg(H_0'+\frac{G_0't}{2\sqrt{\kappa +1}}\bigg) -\frac{G_0(\kappa +2-t^2)}{2\sqrt{\kappa +1-t^2}} \\&-\frac{G_0\hbar ^2}{2\sqrt{\kappa +1}\sqrt{\kappa +1-t^2}(\sqrt{\kappa +1}+\sqrt{\kappa +1-t^2})}, \\ b_{223}=&\frac{\kappa -t^2}{\sqrt{\kappa +1-t^2}}\bigg(H_0+\frac{G_0t}{2\sqrt{\kappa +1}}\bigg) -\frac{G_0t(\kappa -t^2)}{\kappa +1-t^2}-\frac{G_0t(\kappa +2-t^2)}{2(\kappa +1-t^2)}\\&-\frac{G_0t(1-t^2)}{2\sqrt{\kappa +1}(\sqrt{\kappa +1}+\sqrt{\kappa +1-t^2})}, \\ b_{224}=&\frac{G_0(2\kappa +1-t^2)}{\sqrt{\kappa +1-t^2}}+(\kappa +1)(2a_1-\Psi _0),\\ b_{225}=&{\widetilde{F}}\frac{1}{2\sqrt{\kappa +1-t^2}}-{\widetilde{G}}^2\frac{t(\kappa -t^2)}{2(\kappa +1-t^2)} +{\widetilde{G}}{\widetilde{H}}\frac{\kappa -t^2}{\sqrt{\kappa +1-t^2}}, \\ b_{226}=&-\frac{G_0\Psi _0t(\kappa +2-t^2)}{2\sqrt{\kappa +1-t^2}}-\frac{G_0\Psi _0\hbar ^2t}{2\sqrt{\kappa +1}\sqrt{\kappa +1-t^2}(\sqrt{\kappa +1} +\sqrt{\kappa +1-t^2})} \\&+\frac{F_0+\frac{(1+\kappa G_0^2)t}{\sqrt{\kappa +1}}}{2\sqrt{\kappa +1-t^2}}-\Psi _1\bigg(H_0'+\frac{G_0't}{2\sqrt{\kappa +1}}\bigg)\\&+\frac{G_0(2\kappa +1-t^2)}{\sqrt{\kappa +1-t^2}}\bigg(H_0+\frac{G_0t}{2\sqrt{\kappa +1}}\bigg) +(2a_1-\Psi _0)(\kappa +1)\bigg(H_0+\frac{G_0t}{2\sqrt{\kappa +1}}\bigg)\\&-\frac{G_0^2t(\kappa -t^2)}{2(\kappa +1-t^2)}. \end{aligned}$$
The expressions of \(b_{3j}\ (j=1,2,3)\) are
$$\begin{aligned} b_{31}&=\frac{(\kappa +1)(\kappa {\widetilde{G}}+2\kappa G_0)}{\hbar ^2\sqrt{\kappa +1-t^2}},\quad b_{33}=-\frac{(\kappa +1)({\widetilde{F}}+F_0+\frac{(1+\kappa G_0^2)t}{\sqrt{\kappa +1}})}{\hbar ^2},\\ b_{32}&=-\frac{1}{\hbar ^2}\big (b_{321}{\widetilde{X}}+b_{322}{\widetilde{Y}} +b_{323}{\widetilde{G}}+b_{324} +b_{325} +b_{326} +b_{327} \big ), \end{aligned}$$
where
$$\begin{aligned} b_{321}=&\sqrt{1-t^2}\bigg(F_0'+\frac{2\kappa G_0G_0't}{\sqrt{\kappa +1}}\bigg) -\frac{(1+\kappa G_0^2)\hbar ^2}{\sqrt{\kappa +1}\sqrt{\kappa +1-t^2} (\sqrt{\kappa +1}+\sqrt{\kappa +1-t^2})} \\&-\frac{(1+\kappa G_0^2)(\kappa +2-t^2)}{\sqrt{\kappa +1-t^2}}, \\ b_{322}=&-\sqrt{1-t^2}\bigg(F_0'+\frac{2\kappa G_0G_0't}{\sqrt{\kappa +1}}\bigg) -\frac{(1+\kappa G_0^2)(\kappa +2-t^2)}{\sqrt{\kappa +1-t^2}} \\&-\frac{(1+\kappa G_0^2)\hbar ^2}{\sqrt{\kappa +1}\sqrt{\kappa +1-t^2}(\sqrt{\kappa +1}+\sqrt{\kappa +1-t^2})}, \\ b_{323}=&\frac{3\kappa +1-2t^2}{\sqrt{\kappa +1-t^2}}\bigg(F_0+\frac{(1+\kappa G_0^2)t}{\sqrt{\kappa +1}}\bigg) -\frac{4\kappa G_0(1-t^2)}{\sqrt{\kappa +1-t^2}}\bigg(H_0+\frac{G_0 t}{2\sqrt{\kappa +1}}\bigg) \\& -\frac{t(\kappa -t^2)}{\kappa +1-t^2}(1+3\kappa G_0^2)-\frac{(1+\kappa G_0^2)t(\kappa +2-t^2)}{\kappa +1-t^2} -\frac{(1+\kappa G_0^2)t(1-t^2)}{\sqrt{\kappa +1}(\sqrt{\kappa +1}+\sqrt{\kappa +1-t^2})}, \\ b_{324}=&-{\widetilde{H}}\cdot \frac{2(1-t^2)(1+\kappa G_0^2)}{\sqrt{\kappa +1-t^2}} +{\widetilde{F}}\bigg (\frac{2G_0(2\kappa +1-t^2)}{\sqrt{\kappa +1-t^2}}+(\kappa +1)(2a_1-\Psi _0)\bigg ), \\ b_{225}=&-{\widetilde{G}}^2\bigg (\frac{3\kappa G_0t(\kappa -t^2)}{\kappa +1-t^2}+\frac{2\kappa (1-t^2)}{\sqrt{\kappa +1-t^2}}\bigg(H_0+\frac{G_0t}{2\sqrt{\kappa +1}}\bigg)\bigg )\\&-{\widetilde{G}}{\widetilde{H}}\frac{4\kappa G_0(1-t^2)}{\sqrt{\kappa +1-t^2}} +{\widetilde{G}}{\widetilde{F}}\frac{3\kappa +1-2t^2}{\sqrt{\kappa +1-t^2}}, \\ b_{226}=&-{\widetilde{G}}^3\frac{\kappa t(\kappa -t^2)}{\kappa +1-t^2}-{\widetilde{G}}^2{\widetilde{H}}\frac{2\kappa (1-t^2)}{\sqrt{\kappa +1-t^2}}, \\ b_{227}=&-\Psi _1(F_0'+\frac{2\kappa G_0G_0't}{\sqrt{\kappa +1}})+\frac{2G_0(2\kappa +1-t^2)}{\sqrt{\kappa +1-t^2}}(F_0+\frac{(1+\kappa G_0^2)t}{\sqrt{\kappa +1}}) \\& +(2a_1-\Psi _0)(\kappa +1)\bigg(F_0+\frac{(1+\kappa G_0^2)t}{\sqrt{\kappa +1}}\bigg)-\frac{G_0t(1+\kappa G_0^2)(\kappa -t^2)}{\kappa +1-t^2}\\&-\frac{(1+\kappa G_0^2)(\kappa +2-t^2)\Psi _0t}{\sqrt{\kappa +1-t^2}} \\& -\frac{2(1-t^2)(1+\kappa G_0^2)}{\sqrt{\kappa +1-t^2}}\bigg(H_0+\frac{G_0t}{2\sqrt{\kappa +1}}\bigg)\\&-\frac{(1+\kappa G_0^2)\hbar ^2\Psi _0t}{\sqrt{\kappa +1}\sqrt{\kappa +1-t^2}(\sqrt{\kappa +1}+\sqrt{\kappa +1-t^2})}. \end{aligned}$$
The expressions of \(b_{4j}\ (j=1,2,3)\) are
$$\begin{aligned} b_{41}=&-\frac{1}{(1-t^2)({\widetilde{Y}}-{\widetilde{H}}+\frac{{\widetilde{G}}t}{\sqrt{\kappa +1-t^2}}+\Omega _0)}, \\ b_{42}=&-\frac{2a_1+\frac{G_0}{2\sqrt{\kappa +1-t^2}}}{(1-t^2)({\widetilde{Y}}-{\widetilde{H}} +\frac{{\widetilde{G}}t}{\sqrt{\kappa +1-t^2}}+\Omega _0)} \\&-\frac{(2t^2-1)G_0+(2t^2+1) {\widetilde{G}}}{4(1-t^2)\sqrt{\kappa +1-t^2}({\widetilde{Y}}-{\widetilde{H}}+\frac{{\widetilde{G}}t}{\sqrt{\kappa +1-t^2}}+\Omega _0)} +\frac{t}{2(1-t^2)},\\ b_{43}=&-\frac{1}{2\hbar ^2({\widetilde{Y}}-{\widetilde{H}}+\frac{{\widetilde{G}}t}{\sqrt{\kappa +1-t^2}}+\Omega _0)}\big (b_{431} {\widetilde{X}} +b_{432}{\widetilde{Y}} +b_{433}{\widetilde{G}} +b_{434}{\widetilde{H}} +b_{435} +b_{436} \big ), \end{aligned}$$
where
$$\begin{aligned} b_{431}=&(2t^2-1)(a_0+a_1t)+2(a_1t-a_0) +G_0t\sqrt{\kappa +1-t^2} -2(\kappa +1-t^2)(a_0+a_1t), \\ b_{432}=&(2t^2-1)(a_1t-a_0)-2(\kappa +1-t^2) \Phi _0+2a_1t(\kappa +1-t^2) -2t\sqrt{1-t^2}(a_1't-a_0'), \\ b_{433}=&t\sqrt{\kappa +1-t^2}(a_1t-a_0) +(2t^2+1)a_1\sqrt{\kappa +1-t^2}-G_0 \\&-(a_1't-a_0')\hbar +2a_1t\sqrt{\kappa +1-t^2}, \\ b_{434}=&-2(\kappa +1-t^2)(a_0+a_1t)-2a_1t(\kappa +1-t^2), \\ b_{435}=&{\widetilde{X}}^2-\frac{1}{2}{\widetilde{G}}^2+[(2t^2-1)-2(\kappa +1-t^2)]{\widetilde{X}}{\widetilde{Y}} +t\sqrt{\kappa +1-t^2}{\widetilde{X}}{\widetilde{G}}-2(\kappa +1-t^2){\widetilde{Y}}{\widetilde{H}}, \\ b_{436}=&(2t^2-1)(a_1t-a_0)(a_1t+a_0)+(a_1t-a_0)^2+G_0t\sqrt{\kappa +1-t^2}(a_1t-a_0)-\frac{G_0}{2} \\&+\frac{1}{2} -2(\kappa +1-t^2)(a_1t+a_0)\Phi _0+(2t^2-1)G_0a_1\sqrt{\kappa +1-t^2}-(a_1't-a_0')\Omega _1 \\&+ 2a_1(\kappa +1-t^2) \bigg (2a_1-\frac{G_0}{\sqrt{\kappa +1-t^2}} \bigg )+2a_1t(\kappa +1-t^2)\Omega _0. \end{aligned}$$
The expressions of \(b_{5j}\ (j=1,2,3)\) are
$$\begin{aligned} b_{51}=&-\frac{1}{(1-t^2)({\widetilde{X}}+{\widetilde{H}}+\Phi _0)}, \\ b_{52}=&-\frac{2a_1}{(1-t^2)({\widetilde{X}}+{\widetilde{H}} +\Phi _0)}-\frac{(2t^2+1)(G_0+{\widetilde{G}})}{4(1-t^2)\sqrt{\kappa +1-t^2}({\widetilde{X}}+{\widetilde{H}} +\Phi _0)}+\frac{t}{2(1-t^2)}, \\ b_{53}=&-\frac{1}{2\hbar ^2({\widetilde{X}}+{\widetilde{H}}+\Phi _0)}\big (b_{531}{\widetilde{X}} +b_{532}{\widetilde{Y}} +b_{533}{\widetilde{G}} +b_{534}{\widetilde{H}} +b_{535} +b_{536}\big ), \end{aligned}$$
where
$$\begin{aligned} b_{531}&=2t\sqrt{1-t^2} (a_0'+a_1't)+(2t^2-1)(a_0+a_1t) -2G_0t\sqrt{\kappa +1-t^2} \\&\quad +2(\kappa +1-t^2)\bigg(\Omega _0-\frac{G_0t}{\sqrt{\kappa +1-t^2}}\bigg)-2a_1t(\kappa +1-t^2), \\ b_{532}&=(2t^2-1)(a_1t-a_0)+2(a_0+a_1t) +G_0t\sqrt{\kappa +1-t^2}-2(\kappa +1-t^2)(a_1t-a_0), \\ b_{533}&=\hbar (a_0'+a_1't) +t\sqrt{\kappa +1-t^2}(a_0+a_1t)-G_0 \\&\quad +(2t^2+1)a_1\sqrt{\kappa +1-t^2} -2t\sqrt{\kappa +1-t^2}(a_1t-a_0), \\ b_{534}&=2(\kappa +1-t^2)(a_1t-a_0) -2a_1t(\kappa +1-t^2), \\ b_{535}&=-\frac{1}{2}{\widetilde{G}}^2+{\widetilde{X}}{\widetilde{Y}} [(2t^2-1)-2(\kappa +1-t^2)] -{\widetilde{X}}{\widetilde{G}}\cdot 2t\sqrt{\kappa +1-t^2} \\&\quad +{\widetilde{X}}{\widetilde{H}}\cdot 2(\kappa +1-t^2)+{\widetilde{Y}}{\widetilde{G}}\cdot t\sqrt{\kappa +1-t^2}, \\ b_{536}&=(2t^2-1)(a_1t-a_0)(a_1t+a_0)+(a_1t+a_0)^2 +G_0t\sqrt{\kappa +1-t^2}(a_1t+a_0)-\frac{G_0^2}{2} \\&\quad +\frac{1}{2}-2(\kappa +1-t^2)\bigg(\Omega _0-\frac{G_0t}{\sqrt{\kappa +1-t^2}}\bigg)(a_1t-a_0) +a_1G_0\sqrt{\kappa +1-t^2}(2t^2+1) \\&\quad -2G_0t\sqrt{\kappa +1-t^2}(a_1t-a_0)+4a_1^2(\kappa +1-t^2)
-2a_1t(\kappa +1-t^2)\Phi _0+(a_0'+a_1't)\Phi _1. \end{aligned}$$
