  [abridged] We present new scattered light and millimeter observations of the
protoplanetary disk around LkH$\alpha\,330$, using SPHERE/VLT and ALMA,
respectively. The scattered-light SPHERE observations reveal an asymmetric ring
at around 45au from the star in addition to two spiral arms with similar radial
launching points at around 90au. The millimeter observations from ALMA
(resolution of 0.06''$\times$0.04'') mainly show an asymmetric ring located at
110au from the star. In addition to this asymmetry, there are two faint
symmetric rings at 60au and 200au. The $^{12}$CO, $^{13}$CO, and C$^{18}$O
lines seem to be less abundant in the inner disk (these observations have a
resolution of 0.16''$\times$0.11''). The $^{13}$CO peaks at a location similar
to the inner ring observed with SPHERE, suggesting that this line is optically
thick and traces variations of disk temperature instead of gas surface-density
variations, while the C$^{18}$O peaks slightly further away at around 60au. We
compare our observations with hydrodynamical simulations that include gas and
dust evolution, and conclude that a 10$M_{\rm{Jup}}$ mass planet at 60au and in
an eccentric orbit ($e=0.1$) can qualitatively explain most of the observed
structures. A planet in a circular orbit leads to a much narrower concentration
in the millimeter emission, while a planet in a more eccentric orbit leads to a
very eccentric cavity as well. In addition, the outer spiral arm launched by
the planet changes its pitch angle along the spiral due to the eccentricity and
when it interacts with the vortex, potentially appearing in observations as two
distinct spirals. Our observations and models show that LkH$\alpha\,330$ is an
interesting target to search for (eccentric-) planets while they are still
embedded in their parental disk, making it an excellent candidate for studies
on planet-disk interaction.
