Classification des déploiements de germes de systèmes microdifférentiels holonomes de multiplicité 2. (Classification of unfoldings of germs of holonomic microdifferential systems of multiplicity 2) (Q1080994)

Let \(M_ 0\) be a germ of holonomic microdifferential systems of one variable of multiplicity 2 defined on a neighborhood of a point \((0,dt)\in T^*{\mathbb{C}}\). By \textit{B. Malgrange} [Prog. Math. 37, 353-379 (1983; Zbl 0528.32016)], \(M_ 0\) is represented by the form \(M_ 0: tu_ 0=(A_ 0+A_ 1D_ t^{-1})u_ 0,\) where \(A_ 0\) and \(A_ 1\) are constant matrices of order 2 and \(A_ 1\) is of the Jordan form. The author classifies all the stable deployments of such a \(M_ 0\). If \(A_ 0\neq 0\), the system has a versal deployment. If \(A_ 0=0\) and if \(A_ 1\) is of type \(\left( \begin{matrix} \lambda \quad 1\\ 0\quad \lambda \end{matrix} \right)\), the system is stable. When \(A_ 0=0\) and \(A_ 1\) is of type \(\left( \begin{matrix} \lambda_ 1\\ 0\end{matrix} \begin{matrix} 0\\ \lambda_ 2\end{matrix} \right)\), a necessary and sufficient condition for the stability of a deployment of \(M_ 0\) is given.