Consider the following system of first order nonlinear partial differential equations.
$0 = B_y (\frac{\partial B_y}{\partial x} - \frac{\partial B_x}{\partial y}) + B_z (\frac{\partial B_z}{\partial x} - \frac{\partial B_x}{\partial z})$
$0 = B_x (\frac{\partial B_x}{\partial y} - \frac{\partial B_y}{\partial x}) + B_z (\frac{\partial B_z}{\partial y} - \frac{\partial B_y}{\partial z})$
$0 = B_x (\frac{\partial B_x}{\partial z} - \frac{\partial B_z}{\partial x}) + B_y (\frac{\partial B_y}{\partial z} - \frac{\partial B_z}{\partial y})$
$0 = \frac{\partial B_x}{\partial x} + \frac{\partial B_y}{\partial y} + \frac{\partial B_z}{\partial z}$
where $B_x$, $B_y$ and $B_z$ vanish at infinity. Do nontrivial solutions for $B_x$, $B_y$ and $B_z$ exist? Also I think that of the first three equations we can toss out one since it can be derived from the other two.
There are nontrivial solutions with $\nabla\times B\neq 0$. In the literature, $B$ is called a Beltrami vector field. This system arises in the equations of incompressible fluid mechanics, where $B$ there represents the velocity vector field of the fluid. See here for examples of solutions. Other examples are easily found using the Fourier or Laplace transforms, for example, or simply separation of variables.
Note that if $\nabla\times B\neq 0$, then boundary value problems are difficult to solve, since the corresponding solutions $B$ to Helmholtz's equation $\nabla^2B=-\lambda^2 B$ (see here) must also satisfy $\nabla\times B=\lambda B$. However, in the case $\nabla\times B=0$, we get Laplace's equation $\nabla^2B=0$, which is known as potential flow.