  There is a Rota-Baxter algebra structure on the field $A=\mathbf{k}((t))$
with $ P$ being the projection map $A=\mathbf{k}[[t]]\oplus
t^{-1}\mathbf{k}[t^{-1}]$ onto $ \mathbf{k}[[ t]]$. We study the representation
theory and regular-singular decompositions of any finite dimensional $A$-vector
space. The main result shows that the category of finite dimensional
representations is semisimple and consists of exactly three isomorphism classes
of irreducible representations which are all one-dimensional. As a consequence,
the number of $GL_A(V)$-orbits in the set of all regular-singular
decompositions of an $n$-dimensional $ A$-vector space $V$ is $(n+2)(n+1)/2$.
We also use the result to compute the generalized class number, i.e., the
number of the $GL_n(A)$-isomorphism classes of finitely generated
$\mathbf{k}[[t]]$-submodules of $A^n$.
