  A vector field is called a Beltrami vector field, if $B\times(\nabla\times
B)=0$. In this paper we construct 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 such that both have an
orientation-preserving icosahedral symmetry. Both of them have an additional
symmetry with respect to a non-trivial automorphism of the number field
$\mathbb{Q}(\,\sqrt{5}\,)$.
