  In this paper, we use Galois descent techniques to find suitable
representatives of the regular simple representations of the species of type
$(2,2)$ over $k_n := k[\varepsilon^{1/n}]$, where $n$ is a positive integer and
$k:=\mathbb{C}(\!(\varepsilon)\!)$ is the field of Laurent series over the
complexes. These regular representations are essential for the definition of
canonical algebras. Our work is inspired by the work done for species of type
$(1,4)$ on $k$ in ``A model for the canonical algebras of bimodules type (1, 4)
over truncated polynomial rings''. We presents all the regular simple
representations on the $n$-crown quiver, and from these, we establish a partial
classification of regular simple representations of bimodules type $(2,2)$.
