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