## Abstract

Symmetries of the defocusing nonlinear Schrödinger equation are expressed in action-angle coordinates and characterized in terms of the periodic and Dirichlet spectrum of the associated Zakharov-Shabat system. Application: proof of the conjecture that the periodic spectrum ⋯<λ k - ≤λ k + <λ k+1 - ≤⋯ of a Zakharov-Shabat operator is symmetric, i.e. λ k ± =-λ -k ∓ for all k, if and only if the sequence (γ k ) k∈ℤ of gap lengths, γ k :=λ k + -λ k - , is symmetric with respect to k=0.