Integrable deformations of the Dirac--sinh-Gordon system

We construct a two-dimensional family of integrable coupled Dirac–scalar field theories in $1+1$ dimensions, parameterized by $(\text{thz},\alpha )\in [0,\pi \mathrm{/}2{]}^2$, whose Lax connection takes values in $\text{slC}$ throughout. The family arises as the orbit of the Dirac–sinh-Gordon system under the $U(1)\times U(1)$ maximal torus of $\mathrm{I}\mathrm{n}\mathrm{n}(\text{slC})\cong PSL(2,\text{CC})$: the $\text{thz}$-$U(1)$ rotates the constant phase of the Dirac mass; the $\alpha$-$U(1)$ rotates its field-dependent phase via $\beta \to \mathrm{\mid }\beta \mathrm{\mid }{\text{ee}}^{\text{im}\alpha }$. Integrability throughout the parameter space follows from a single principle: any automorphism of the Lax algebra preserves the zero-curvature condition, since the condition depends only on the Lie bracket of $\text{slC}$.