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We concisely summarize a method of finding all rational solutions to aninhomogeneous rational ODE system of arbitrary order (but solvable for itshighest order terms) by converting it into a finite dimensional linear algebraproblem. This method is then used to solve the problem of conclusively decidingwhen certain rational ODE systems in upper triangular form can or cannot bereduced to diagonal form by differential operators with rational coefficients.As specific examples, we consider systems of coupled Regge-Wheeler equations,which have naturally appeared in previous work on vector and tensorperturbations on the Schwarzschild black hole spacetime. Our systematicapproach reproduces and complements identities that have been previously foundby trial and error methods.

Reducing triangular systems of ODEs with rational coefficients, with applications to coupled Regge-Wheeler equations