Triangularisation of Singularly Perturbed Linear Systems of Ordinary Differential Equations: Rank Two at Regular Singular Points

We study singularly perturbed linear systems of rank two of ordinary differential equations of the form $varepsilon xpartial_x psi (x, varepsilon) + A (x, varepsilon) psi (x, varepsilon) = 0$, with a regular singularity at $x = 0$, and with a fixed asymptotic regularity in the perturbation parameter $varepsilon$ of Gevrey type in a fixed sector. We show that such systems can be put into an upper-triangular form by means of holomorphic gauge transformations which are also Gevrey in the perturbation parameter $varepsilon$ in the same sector. We use this result to construct a family in $varepsilon$ of Levelt filtrations which specialise to the usual Levelt filtration for every fixed nonzero value of $varepsilon$; this family of filtrations recovers in the $varepsilon to 0$ limit the eigen-decomposition for the $varepsilon$-leading-order of the matrix $A (x, varepsilon)$, and also recovers in the $x to 0$ limit the eigen-decomposition of the residue matrix $A (0, varepsilon)$.