In $\mathcal {D}$-modules theory, Gauss-Manin systems are defined by the direct image of the structure sheaf $\mathcal {O}$ by a morphism. A major theorem says that these systems have only regular singularities. This paper examines the irregularity of an analogue of the Gauss-Manin systems. It consists in the direct image complex $f_+(\mathcal {O}\rme ^g)$ of a $\mathcal {D}$-module twisted by the exponential of a polynomial $g$ by another polynomial $f$, where $f$ and $g$ are two polynomials in two variables. The analogue of the Gauss-Manin systems can have irregular singularities (at finite distance and at infinity). We express an invariant associated with the irregularity of these systems at $c\in \mathbb {P}^1$ by the geometry of the map $(f,g)$.