  We construct classes of homogeneous random fields on a three-dimensional
Euclidean space that take values in linear spaces of tensors of a fixed rank
and are isotropic with respect to a fixed orthogonal representation of the
group of $3\times 3$ orthogonal matrices. The constructed classes depend on
finitely many isotropic spectral densities. We say that such a field belong to
either the Mat\'{e}rn or the dual Mat\'{e}rn class if all of the above
densities are Mat\'{e}rn or dual Mat\'{e}rn. Several examples are considered.
