  We investigate the relation between the dynamical features of a supercooled
liquid and those of its potential energy landscape, focusing on a model liquid
with density anomalies. We consider, at fixed temperature, pairs of state
points with different density but the same diffusion constant, and find that
surprisingly they have identical dynamical features at all length and time
scales. This is shown by the collapse of their mean square displacements and of
their self--intermediate scattering functions at different wavevectors. We then
investigate how the features of the energy landscape change with density, and
establish that state points with equal diffusion constant have different
landscapes. In particular, we find a correlation between the fraction of
instantaneous normal modes connecting different energy minima and the diffusion
constant, but unlike in other systems these two quantities are not in
one--to--one correspondence with each other, showing that additional landscape
features must be relevant in determining the diffusion constant.
