  A 3-dimensional vector field $B$ is said to be Beltrami vector field (force
free-magnetic vector field in physics), if $B\times(\nabla\times B)=0$.
Motivated by our investigations on projective an polynomial superflows, and as
an important side result, in the first paper on this topic we constructed two
unique Beltrami vector fields $\mathfrak{I}$ and $\mathfrak{Y}$, such that
$\nabla\times\mathfrak{I}=\mathfrak{I}$,
$\nabla\times\mathfrak{Y}=\mathfrak{Y}$, and that both have
orientation-preserving icosahedral symmetry (group of order $60$). In the
current paper we extend these results to the tetrahedral and octahedral cases,
and (together with an icosahedral case) we calculate all simplest Beltrami
fields with polyhedral symmetries arising from solutions to the Helmholtz
equation of any order (the first aforementioned paper being an order 1
approach). The notion of Beltrami vector field, slightly relaxed, generalizes
to any dimension. In this paper we also present 2-dimensional vector fields
which have a dihedral symmetry $\mathbb{D}_{2d+1}$ of order $4d+2$. A much more
detailed analysis is carried out in case $d=1$. One of these fields is
particularly exceptional since it is the only case in our investigations which
arises from the order $0$ approach to the Helmholtz equation, thus relating
this flow to the $ABC$ flow.
