Nonlinear PT-symmetric model and integrability I am coming from a maths background and these concepts are a bit new to me. I am studying an exercise paper and I am encountering the following problem. With $u_i(t)=u_i$ and of course $u_i u_i^*=|u|^2$ for ($i=1,2$) where $\chi$, $\beta$,$\omega_0$ and $\gamma$ are constants, we have the system:
\begin{equation}  
\begin{split}
i \dot{u_1}  & = -(\omega_0 + i \beta) u_1 + \chi |u_1|^2 u_1 + \gamma u_2  \\
i \dot{u_2}  & = -(\omega_0 - i \beta) u_2 + \chi |u_2|^2 u_2 + \gamma u_1.
\end{split}
\end{equation}
*Short physical insight about the problem: It is a PT-symmetric nonlinear dimer. The system consists of two PT -coupled waveguide elements with Kerr nonlinearity of strength $\chi$ which operate at a a frequency $\omega_0$. Each of the
waveguides is singlemoded, one providing gain and the other an equal amount of loss. (Where PT is the combined parity-time transformation,  $u_1 \leftrightarrow u_2$ (parity transformation) and $t \to -t$ (time reversal)).
He wants to identify the two conserved quantities for $\beta=0$ and $\beta\neq0$  respectively, also write the system in a form of $S_i$'s. The Stokes parameters are the following:
\begin{equation}  
\begin{split}
& S_0 = |u_1|^2 + |u_2|^2,\quad
S_3 = |u_1|^2 - |u_2|^2,\\
& S_1 =  u_1 u_2^* + u_1^* u_2,\quad
S_2 = i(u_1 u_2^* - u_1^* u_2).
\end{split}
\end{equation}
Here it is what I did (I wrote the important stuff here, and not many of the calculations):
A) For $\beta=0$, power $N$ is conserved which is easy by taking the time derivative:
\begin{equation}  
N = S_0=|u_1|^2 + |u_2|^2 \\
\end{equation}
\begin{equation} 
\begin{split}
\frac{\mathrm{d}N}{\mathrm{d} t} = \frac{\mathrm{d} }{\mathrm{d} t}(|u_1|^2+|u_2|^2)\Rightarrow  \\
\frac{\mathrm{d}N}{\mathrm{d} t}=(\dot{u_1}u_1^*+u_1\dot{u_1}^*)+(\dot{u_2}u_2^*+u_2\dot{u_2}^*) \Rightarrow  \\
\frac{\mathrm{d}N}{\mathrm{d} t}=0
\end{split}
\end{equation}
Not sure what kind of energy $E$ is conserved. 
B) For $\beta\neq0$ I can't find any conserved quantities.
I also wrote the initial system in a different way by multiplying it with $u_1^*$ and $u_2^*$ and then adding them together in an effort to write then into a form of $S_i$'s and get something useful  but I got nowhere.
So, any ideas or help in solving the system or finding the conserved qunatities would be appreciated because I can't find much information about this topic online.
 A: *

*OP is studying a coupled system of the form
$$ \begin{split}
i \dot{u}_1  & ~=~ - i\beta u_1 + \chi |u_1|^2 u_1 + \gamma u_2, \qquad  u_1,u_2~\in~\mathbb{C}, \cr
i \dot{u}_2  & ~=~ + i\beta^{\ast} u_2 + \chi |u_2|^2 u_2 + \gamma^{\ast}u_1,\qquad \chi~\in~\mathbb{R}\backslash\{0\},\qquad \beta~\in~\mathbb{C},\qquad \gamma~\in~\mathbb{C}\backslash\{0\}. 
\end{split} \tag{A} $$

*This system (A) is $PT$-symmetric. The time-reflection symmetry is $t \leftrightarrow -t$. The parity symmetry $P$ is $u_1 \leftrightarrow u_2^{\ast}$. 

*The system (A) has a Lagrangian formulation for ${\rm Re}\beta=0$:
$$ L~:=~i \sum_{k=1}^2 u^{\ast}_k\dot{u}_k  - H, \qquad 
H~:=~ 2{\rm Re}(\gamma u^{\ast}_1 u_2)
+\sum_{k=1}^2\left(|u_k|^2{\rm Im} \beta
+\frac{\chi}{2}|u_k|^4\right).\tag{B} $$ 

*By redefinition of 
$$u_k~\longrightarrow~ e^{-it{\rm Im}\beta}u_k, \qquad k~\in~\{1,2\},\tag{C}$$ 
we can (and will) assume that $\beta\in\mathbb{R}$ is real. By redefinition of $u_1$ and $u_2$ with opposite phases we can (and will) assume that $\gamma\in\mathbb{R}$ is real.

*If we define polar coordinates $u_k=r_k e^{i\theta_k}$, the equations of motion (A) become
$$ \begin{split}
\dot{r}_1&~=~-\beta r_1 +\gamma r_2\sin(\theta_2\!-\!\theta_1), \cr
\dot{r}_2&~=~\beta r_2 -\gamma r_1\sin(\theta_2\!-\!\theta_1), \cr
-\dot{\theta}_1
&~=~\chi r_1^2 +\gamma \frac{r_2}{r_1}\cos(\theta_2\!-\!\theta_1), \cr
-\dot{\theta}_2
&~=~\chi r_2^2 +\gamma \frac{r_1}{r_2}\cos(\theta_2\!-\!\theta_1).
\end{split}\tag{D} $$

*The Stokes parameters $S_{\mu}:={\bf u}^{\dagger}\sigma_{\mu}{\bf u}$ are given by
$$ \begin{split} 
S_0&~=~r_1^2+r_2^2, \cr
S_3&~=~r_1^2-r_2^2, \cr
S_1&~=~2r_1r_2\cos(\theta_2\!-\!\theta_1),\cr
S_2&~=~2r_1r_2\sin(\theta_2\!-\!\theta_1).
\end{split}\tag{E} $$
They satisfy a constraint $\vec{S}^2=S^2_0$. The equations of motion (D) become
$$\begin{split}
\dot{S}_0  &~=~-2\beta S_3, \cr
\dot{S}_3  &~=~-2\beta S_0+2\gamma S_2, \cr
\dot{S}_1  &~=~-\chi S_2 S_3, \cr
\dot{S}_2  &~=~(\chi S_1 -2\gamma) S_3.
\end{split}\tag{F} $$

*Eq. (5) in Ref. 1 is eq. (A) with $\gamma=1$. There are listed 2 conserved quantities $C$ and $J$ in eq. (7) of Ref. 1, cf. above comment by OP. We guess that the pertinent generalizations read
$$ \begin{split}
C^2~&:=~(\chi S_1-2\gamma)^2+(\chi S_2)^2, \cr 
J~&:=~S_0+\frac{2\beta}{\chi}\arcsin\frac{\chi S_1-2\gamma}{C}.
\end{split}\tag{G}$$
It is straightforward to check that they are indeed constants of motion. In other words, the system (A) with a 4-dimensional real phase space is integrable. 

*Afterthought: It is tempting to define 
$$\Sigma_{\pm}~:=~\left(S_1 -\frac{2\gamma}{\chi}\right)\pm i S_2~=~\Sigma_{\mp}^{\ast}.\tag{H}$$ Then 
$$\dot{\Sigma}_{\pm}~=~\pm i \chi S_3\Sigma_{\pm},\tag{I}$$ and then
$$Q_{\pm}~:=~\exp\left\{\frac{i\chi S_0}{2\beta}\right\}\Sigma_{\pm}~=~Q_{\mp}^{\ast}\tag{J}$$ 
is 1 complex (corresponding to 2 real) constants of motion.
References:


*

*H. Ramezani, T. Kottos, R. El-Ganainy & D.N. Christodoulides, Unidirectional Nonlinear PT-symmetric Optical Structures, arXiv:1005.5189.

