We consider Lindbladian operator dynamics in many-body quantum systems with one or more integrals of motion (IOMs), subject to weak local dissipation. We demonstrate that IOMs with small support become slow modes of these dynamics, in the sense that their Frobenius norms decay more slowly compared to generic operators. As a result, the eigenoperators of such Lindbladians with slowest decay rates have a large overlap with the IOMs of the underlying Hamiltonian. We demonstrate this correspondence between slow modes and IOMs numerically for a number of many-body models and further corroborate it with perturbative arguments. These results open up another method for the identification of IOMs and provide insights into the dissipative many-body dynamics.

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• Dissipative dynamics
• Open quantum systems
• Integrable systems
• Lindblad equation