  The Fueter theorem provides a two step procedure to build an axially
monogenic function, i.e. a null-solutions of the Cauchy-Riemann operator in $
\mathbb{R}^4$, denoted by $ \mathcal{D}$. In the first step a holomorphic
function is extended to a slice hyperholomorphic function, by means of the
so-called slice operator. In the second step a monogenic function is built by
applying the Laplace operator in four real variables ($\Delta$) to the slice
hyperholomorphic function. In this paper we use the factorization of the
Laplace operator, i.e. $\Delta= \mathcal{\overline{D}} \mathcal{D}$ to split
the previous procedure. From this splitting we get a class of functions that
lies between the set of slice hyperholomorphic functions and the set of axially
monogenic functions: the set of axially polyanalytic functions of order 2, i.e.
null-solutions of $ \mathcal{D}^2$. We show an integral representation formula
for this kind of functions. The formula obtained is fundamental to define the
associated functional calculus on the $S$-spectrum. As far as the authors know,
this is the first time that a monogenic polyanalytic functional calculus has
been taken into consideration.
