  On-off intermittency occurs in nonequilibrium physical systems close to
bifurcation points and is characterised by an aperiodic switching between a
large-amplitude "on" state and a small-amplitude "off" state. L\'evy on-off
intermittency is a recently introduced generalisation of on-off intermittency
to multiplicative L\'evy noise, which depends on a stability parameter $\alpha$
and a skewness parameter $\beta$. Here, we derive two novel results on L\'evy
on-off intermittency by leveraging known exact results on the first-passage
time statistics of L\'evy flights. First, we compute anomalous critical
exponents explicitly as a function of arbitrary L\'evy noise parameters
$(\alpha,\beta)$ for the first time, by a heuristic method, complementing
previous results. The predictions are verified using numerical solutions of the
fractional Fokker-Planck equation. Second, we derive the power spectrum $S(f)$
of L\'evy on-off intermittency and show that it displays a power law
$S(f)\propto f^\kappa$ at low frequencies $f$, where $\kappa\in (-1,0)$ depends
on the noise parameters $\alpha,\beta$. An explicit expression for $\kappa$ is
obtained in terms of $(\alpha,\beta)$. The predictions are verified using long
time series realisations of L\'evy on-off intermittency. Our findings help shed
light on instabilities subject to non-equilibrium, power-law-distributed
fluctuations, emphasizing that their properties can differ starkly from the
case of Gaussian fluctuations.
