Let $U(\lambda)$ and $V(\lambda)$ be matrix-valued functions of $(x, y)$ depending on the auxiliary parameter $\lambda$. Consider a system of linear PDEs

$$\frac{\partial}{\partial x} \Phi=U(\lambda) \Phi, \quad \frac{\partial}{\partial y} \Phi=V(\lambda) \Phi$$

where $\Phi$ is a column vector whose components depend on $(x, y, \lambda)$. Derive the zero curvature representation as the compatibility conditions for this system.

Assume that

$$U(\lambda)=-\left(\begin{array}{ccc} u_{x} & 0 & \lambda \\ 1 & -u_{x} & 0 \\ 0 & 1 & 0 \end{array}\right), \quad V(\lambda)=-\left(\begin{array}{ccc} 0 & e^{-2 u} & 0 \\ 0 & 0 & e^{u} \\ \lambda^{-1} e^{u} & 0 & 0 \end{array}\right)$$

and show that (1) is compatible if the function $u=u(x, y)$ satisfies the PDE

$$\frac{\partial^{2} u}{\partial x \partial y}=F(u)$$

for some $F(u)$ which should be determined.

Show that the transformation

$$(x, y) \longrightarrow\left(c x, c^{-1} y\right), \quad c \in \mathbb{R} \backslash\{0\}$$

forms a symmetry group of the PDE (2) and find the vector field generating this group.

Find the ODE characterising the group-invariant solutions of (2).