We discover a family of local deformations that leave part of the spectrum intact for strongly interacting and exactly solvable quantum many-body systems. Since the deformation preserves the Bethe ansatz equations (BAE), it is dubbed the iso-BAE flow. Although all theories on the flow share the same BAE, the spectra are different. Part of the spectrum remains intact along the whole flow. Such states are protected by an emergent symmetry. The remaining parts of the spectrum change continuously along the flow and are doubly degenerate for even-length spin chains. For odd-length chains, the deformed spectrum also comprises doubly degenerate pairs apart from the sector with magnon number $(L+1)/2$, where $L$ is the length of the spin chain. We discuss the iso-BAE flow for the ${\mathrm{XXX}}_{1/2}$ model in detail and show that the iso-BAE flows exist for more general models including $q$-deformed XXZ as well as higher-spin ${\mathrm{XXX}}_s$ spin chains.

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• Quantum statistical mechanics
• Quantum many-body systems
• Bethe ansatz
• Exact solutions for many-body systems
• Heisenberg model
• Integrable systems
• Quantum spin chains