The quest to elucidate potential physical interpretations of the infinitely many symmetries and/or conservation laws inherent in integrable systems constitutes a longstanding challenge within the scientific communities of physics and mathematics. This letter delves into this unresolved issue through an unconventional lens. For a specified integrable system, there exist various categories of n-wave solutions, such as the n-soliton solutions, multiple breathers, complexitons, and the n-periodic wave solutions, wherein n denotes an arbitrary integer that can potentially approach infinity. Each subwave comprising the n-wave solution may possess certain free parameters, including center parameters ci, width parameters ki, and periodic parameters mi. It is evident that these solutions are translationally invariant with respect to all these free parameters. We postulate that the entirety of the recognized infinitely many symmetries merely constitute linear combinations of these finite wave parameter translation symmetries. This conjecture appears to hold true for all integrable systems with n-wave solutions. By considering the renowned KdV equation and the Burgers equation as exemplar cases, the conjecture is substantiated for the n-soliton solutions. It is unequivocal that any linear combination of the wave parameter translation symmetries retains its status as a symmetry associated with the particular solution.
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