Non-linear partial differential equation

Solution 1:

The eigenvectors of the curl operator $$\nabla \times {\bf A}$$ in Cylindrical coordinates are the Bessel functions $J_0( a r)$, you will have to plug back this in to see how $a$ depends on $\lambda$. But all this assumes $\lambda$ is a constant. In your problem, you don't have $\lambda$ constant, but special cases are sometimes useful.

Solution 2:

Unless I'm missing something, the constraint (2)

$\nabla \lambda \times \mathbf{A} = 0$

means that the vector field must be of the form

$\mathbf{A} = f \nabla \lambda$

where f is a scalar function. Inserting into (1) yields

$\nabla f \times \nabla \lambda = f \lambda \nabla \lambda$.

The only possible solutions are $\nabla\lambda = 0$, so $\lambda$ has to be a constant. Together with the other answer about the eigenvectors of the curl operator, maybe this solves your problem.