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We study a reflectionless PT-symmetric quantum system described by the pairof complexified Scarf II potentials mutually displaced in the half of theirpure imaginary period. Analyzing the rich set of intertwining discretesymmetries of the pair, we find an exotic supersymmetric structure based onthree matrix differential operators that encode all the properties of thesystem, including its reflectionless (finite-gap) nature. The structure werevealed particularly sheds new light on the splitting of the discrete statesinto two families, related to the bound and resonance states in Hermitian ScarfII counterpart systems, on which two different series of irreduciblerepresentations of sl(2,C) are realized.

Self-isospectral tri-supersymmetry in PT-symmetric quantum systems with pure imaginary periodicity