  We consider a continuous-time symmetric supercritical branching random walk
on a multidimensional lattice with a finite set of the particle generation
centres, i.e. branching sources. The main object of study is the evolutionary
operator for the mean number of particles both at an arbitrary point and on the
entire lattice. The existence of positive eigenvalues in the spectrum of an
evolutionary operator results in an exponential growth of the number of
particles in branching random walks, called supercritical in the such case. For
supercritical branching random walks, it is shown that the amount of positive
eigenvalues of the evolutionary operator, counting their multiplicity, does not
exceed the amount of branching sources on the lattice, while the maximal of
these eigenvalues is always simple. We demonstrate that the appearance of
multiple lower eigenvalues in the spectrum of the evolutionary operator can be
caused by a kind of `symmetry' in the spatial configuration of branching
sources. The presented results are based on Green's function representation of
transition probabilities of an underlying random walk and cover not only the
case of the finite variance of jumps but also a less studied case of infinite
variance of jumps.
