  We study the classical and quantum ergodic lemon billiard introduced by
Heller and Tomsovic in Phys. Today 46(7), 38 (1993), for the case $B=1/2$,
which is a classically ergodic system (without a rigorous proof) exhibiting
strong stickiness regions around a zero-measure bouncing ball modes. The
structure of the classical stickiness regions is uncovered in the S-plots
introduced by Lozej [Phys. Rev. E 101, 052204 (2020)]. A unique classical
transport or diffusion time cannot be defined. As a consequence the quantum
states are characterized by the following nonuniversal properties: (i) All
eigenstates are chaotic but localized as exhibited in the Poincar\'e-Husimi
(PH) functions. (ii) The entropy localization measure A (also the normalized
inverse participation ratio) has a nonuniversal distribution, typically
bimodal, thus deviating from the beta distribution, the latter one being
characteristic of uniformly chaotic systems with no stickiness regions. (iii)
The energy-level spacing distribution is Berry-Robnik-Brody (BRB), capturing
two effects: the quantally divided phase space (because most of the PH
functions are either the inner-ones or the outer-ones, dictated by the
classical stickiness, with an effective parameter $\mu_1$ measuring the size of
the inner region bordered by the sticky invariant object, namely, a cantorus),
and the localization of PH functions characterized by the level repulsion
(Brody) parameter $\beta$. (iv) In the energy range considered (between 20 000
states to 400 000 states above the ground state) the picture (the structure of
the eigenstates and the statistics of the energy spectra) is not changing
qualitatively, as $\beta$ fluctuates around 0.8, while $\mu_1$ decreases almost
monotonically, with increasing energy.
