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We make two observations on the motion of coupled particles in a periodicpotential. Coupled pendula, or the space-discretized sine-Gordon equation is anexample of this problem. Linearized spectrum of the synchronous motion turnsout to have a hidden asymptotic periodicity in its dependence on the energy;this is the gist of the first observation. Our second observation is thediscovery of a special property of the purely sinusoidal potentials: thelinearization around the synchronous solution is equivalent to the classicalLam\`e equation. As a consequence, {\it all but one instability zones of thelinearized equation collapse to a point for the one-harmonic potentials}. Thisprovides a new example where Lam\'e's finite zone potential arises in thesimplest possible setting.

Spectral asymptotics and Lam\'e spectrum for coupled particles in periodic potentials