Generalized Beltrami fields. Exact solutions

We study generalized Beltrami fields defined as solutions to the system $\operatorname{rot}^n A=\lambda A$, where $\lambda$ is a function and $A=(P,Q,R)$ is a vector-valued function of the variables $(x,y,z)$, $n\in {\Bbb N}$. For $\lambda=1$ and arbitrary natural $n$, the system is reduced to a completely integrable form, with the result depending on the parity of $n$. For $n=1$ and an arbitrary function $\lambda$, the system is also reduced to a completely integrable form.