### A system of differential equations for the Airy process.

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For a Fuchsian system $dY/dx=({\sum}_{j=}^{p}\left({A}_{j}\right)/(x-{t}_{j}))Y$, (F) $t\u2081,t\u2082,...,{t}_{p}$ being distinct points in ℂ and $A\u2081,A\u2082,...,{A}_{p}\in M(n\times n;\u2102)$, the number α of accessory parameters is determined by the spectral types $s\left(A\u2080\right),s\left(A\u2081\right),...,s\left({A}_{p}\right)$, where $A\u2080=-{\sum}_{j=1}^{p}{A}_{j}$. We call the set $z=(z\u2081,z\u2082,...,{z}_{\alpha})$ of α parameters a regular coordinate if all entries of the ${A}_{j}$ are rational functions in z. It is not yet known that, for any irreducibly realizable set of spectral types, a regular coordinate does exist. In this paper we study a process of obtaining a new regular coordinate from a given one by a coalescence of eigenvalues of the matrices...