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

► A two-parametric family of pairs of reflectionless self-isospectral PT-symmetric systems is studied. ► A new exotic supersymmetric structure based on three integrals of motion is found. ► The splitting of the discrete states into two families is explained. ► The role of various discrete symmetries and the imaginary period is understood.