Let $a,b$ two smooth functions from the open square square $I^{2}$ in $\mathbb{R}^{2}$ to $\mathbb{R}^{4}$. In particular, assume $a(t,u)$ and $b(t,u)$ be linearly independent for all $(t,u) \in I^{2}$.
I need to study the PDE problem in $ x \in C^{\infty}(I^{2},\mathbb{R}^{4})$ given by the underdetermined system $$ \begin{cases} \displaystyle \frac{\partial x}{\partial t} \cdot a = 0 \\ \displaystyle \frac{\partial x}{\partial u} \cdot b = 0 \\ \displaystyle \frac{\partial x}{\partial t} \cdot b = \frac{\partial x}{\partial u} \cdot a \end{cases} $$ (summations over $i$ understood) and the auxiliary condition $$x(\cdot,0)= \frac{\partial x}{\partial u}(\cdot,0)=0\,.$$
Question: how would you go about solving this problem, i.e. representing its solution set?
Before somebody legitimately asks for motivation, let me finally say that this problem is linked to the isometric flexibility of a surface in $\mathbb{R}^{4}$.