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 $(\thz,\alpha)\in[0,\pi/2]^2$, whose Lax connection takes values in $\slC$ throughout. The family arises as the orbit of the Dirac--sinh-Gordon system under the $U(1)\times U(1)$ maximal torus of $\mathrm{Inn}(\slC)\cong PSL(2,\CC)$: the $\thz$-$U(1)$ rotates the constant phase of the Dirac mass; the $\alpha$-$U(1)$ rotates its field-dependent phase via $\beta\to|\beta|\ee^{\im\alpha}$. 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 $\slC$.