  The energy spectrum of neutrino-induced upward-going muons in MACRO has been
analysed in terms of relativity principles violating effects, keeping standard
mass-induced atmospheric neutrino oscillations as the dominant source of
$\nu_{\mu} \to \nu_{\tau}$ transitions. The data disfavor these exotic
possibilities even at a sub-dominant level, and stringent 90% C.L. limits are
placed on the Lorentz invariance violation parameter $|\Delta v| < 6 \times
10^{-24}$ at $\sin 2{\theta}_v$ = 0 and $|\Delta v| < 2.5 \div 5 \times
10^{-26}$ at $\sin 2{\theta}_v$ = $\pm$1. These limits can also be
re-interpreted as upper bounds on the parameters describing violation of the
Equivalence Principle.
