  The Schr\"odinger equation for quantum dot lattices with non-cubic,
non-Bravais lattices built up from elliptical dots is investigated. The Coulomb
interaction between the dots is considered in dipole approximation. Then only
the center of mass (c.m.) coordinates of different dots couple with each other.
This c.m. subsystem can be solved exactly and provides magneto- phonon like
collective excitations. The inter-dot interaction is involved only through a
single interaction parameter. The relative coordinates of individual dots form
decoupled subsystems giving rise to intra-dot excitations. As an example, the
latter are calculated exactly for two-electron dots.
  Emphasis is layed on qualitative effects like: i) Influence of the magnetic
field on the lattice instability due to inter-dot interaction, ii) Closing of
the gap between the lower and the upper c.m. mode at B=0 for elliptical dots
due to dot interaction, and iii) Kinks in the single dot excitation energies
(versus magnetic field) due to change of ground state angular momentum. It is
shown that for obtaining striking qualitative effects one should go beyond
simple cubic lattices with spherical dots. We also prove a more general version
of the Kohn Theorem for quantum dot lattices. It is shown that for observing
effects of electron- electron interaction between the dots in FIR spectra
(breaking Kohn's Theorem) one has to consider dot lattices with at least two
dot species with different confinement tensors.
