Certain surfaces in mechanics are endowed with the fundamental forms

\begin{align} \text{I} &= \mathrm{d}u^2+\mathrm{d}v^2+2\cos\gamma\: \mathrm{d}u\: \mathrm{d}v \\ \text{II} &= \alpha\left(\gamma \right)\:\gamma_u\:\mathrm{d}u^2-\beta\left(\gamma \right)\:\gamma_v\:\mathrm{d}v^2+2\tau\:\sin\gamma\: \mathrm{d}u\: \mathrm{d}v, \\ \end{align}

with $(u,v)\in [0,a]^2$, for some $a$ (e.g. 1), and $\alpha$ and $\beta$ the following functions:

\begin{align} \alpha&=\frac{\sin\rho+\sin\sigma\cos\gamma }{\sqrt{\cos^2\sigma\sin^2\gamma-\left(\sin\rho+\sin\sigma\cos\gamma \right)^2}}\\ \beta&=-\frac{\sin\sigma+\sin\rho\cos\gamma }{\sqrt{\cos^2\rho\sin^2\gamma-\left(\sin\sigma+\sin\rho\cos\gamma \right)^2}}. \end{align}

Here $\sigma$ and $\rho$ are fixed constants (between $0$ and $\pi$). The functions $\gamma=\gamma(u,v)$ and $\tau=\tau(u,v)$ must satisfy the Gauss-Codazzi equations, which read

\begin{align} -\frac{1}{\sin\sigma}\left(\tau-\frac{\alpha}{\sin\gamma}\gamma_v \right)_u&=\frac{\gamma_u \gamma_v}{\sqrt{\cos^2\sigma\sin^2\gamma-\left(\sin\rho+\sin\sigma\cos\gamma \right)^2}} \\ -\frac{1}{\sin\rho}\left(\tau+\frac{\beta}{\sin\gamma}\gamma_u \right)_v&=\frac{\gamma_u \gamma_v}{\sqrt{\cos^2\sigma\sin^2\gamma-\left(\sin\rho+\sin\sigma\cos\gamma \right)^2}} \\ \tau^2&=\frac{\gamma_{uv}}{\sin\gamma}-\alpha\beta\:\frac{\gamma_u \gamma_v}{\sin^2\gamma}. \end{align}

The point is to characterize the space of solutions of this overdetermined PDEs system. Since the right hand sides of the first two equations are identical, we introduce some "potential" $\phi$ such that $\frac{1}{\sin\sigma}\left(\tau-\frac{\alpha}{\sin\gamma}\gamma_v \right)=\phi_v$ and $\frac{1}{\sin\rho}\left(\tau+\frac{\beta}{\sin\gamma}\gamma_u \right)=\phi_u$, thus eliminating $\tau$ the system becomes

\begin{align} \phi_{uv}&=-\frac{\gamma_u \gamma_v}{\sqrt{\cos^2\sigma\sin^2\gamma-\left(\sin\rho+\sin\sigma\cos\gamma \right)^2}} \tag{*}\\ \gamma_{uv}&=\sin\sigma\sin\rho\:\phi_u \phi_v\:\sin\gamma-\sin\sigma\:\beta \:\gamma_u\:\phi_v+\sin\rho\:\alpha\:\gamma_v\:\phi_u, \tag{**} \end{align}

with the constraint

\begin{align} \sin \sigma \:\phi_v+\frac{\alpha}{\sin\gamma}\gamma_v=\sin \rho\:\phi_u-\frac{\beta}{\sin\gamma}\gamma_u. \tag{c} \end{align}

Were it not for the constraint, $(*)$ and $(**)$ would set a well posed hyperbolic PDE system from which one can get a unique solution specifying, say, $\gamma(u,0),\:\gamma(0,v)$ and $\phi(u,0),\:\phi(0,v)$.

Is there a standard procedure to analyze systems with lower order constraints like $\{(*),\:(**),\:(\text{c})\}$, how to find particular non-trivial ($\sigma\neq \rho \neq 0$) solutions and how much freedom in boundary/initial conditions does one have?