On rigidity of Pfaffian systems coming from Okubo systems (Q2731419)

Let \(n\) be an integer, \(A\) an \(n\times n\) matrix satisfying NEWLINE\[NEWLINE(A-\rho_1I_n)(A-\rho_2I_n)=ONEWLINE\]NEWLINE for some complex numbers \(\rho_1\), \(\rho_2\) \((\rho_1\neq\rho_2)\), and \(B\) an \(n\times n\) constant diagonal matrix of the form \(B=\lambda_1I_{n_1}\oplus\cdots\oplus \lambda_pI_{n_p}\), where \(\lambda_1,\lambda_2,\dots,\lambda_p\) are mutually distinct complex numbers.NEWLINENEWLINENEWLINEIt is proved that the local system defined by the Pfaffian system NEWLINE\[NEWLINE\begin{aligned} & dZ=\Omega Z,\\ & \Omega=(xI_n-B)^{-1}Adx+(A-(\rho_1+\rho_2)I_n)(yI_n-B)^{-1}dy-A\;\frac{d(x-y)}{x-y}\end{aligned}NEWLINE\]NEWLINE is rigid if and only if the local system defined by the Pfaffian system NEWLINE\[NEWLINE(xI_n-B)\;\frac{dY}{dx}=AYNEWLINE\]NEWLINE is rigid.