  We show that an n-gap periodic quantum system with parity-even smooth
potential admits $2^n-1$ isospectral super-extensions. Each is described by a
tri-supersymmetry that originates from a higher-order differential operator of
the Lax pair and two-term nonsingular decompositions of it; its local part
corresponds to a spontaneously partially broken centrally extended nonlinear
N=4 supersymmetry. We conjecture that any finite-gap system having antiperiodic
singlet states admits a self-isospectral tri-supersymmetric extension with the
partner potential to be the original one translated for a half-period. Applying
the theory to a broad class of finite-gap elliptic systems described by a
two-parametric associated Lame equation, our conjecture is supported by the
explicit construction of the self-isospectral tri-supersymmetric pairs. We find
that the spontaneously broken tri-supersymmetry of the self-isospectral
periodic system is recovered in the infinite period limit.
