  We analyze a continuous and discrete symmetries of the maximum lifetime
problem in two dimensional sensor networks. We show, how a symmetry of the
network and invariance of the problem under a given transformation group $G$
can be utilized to simplify its solution. We prove, that for a $G$-invariant
maximum lifetime problem there exists a $G$-invariant solution. Constrains
which follow from the $G$-invariance allow to reduce the problem and its
solution to a subset, an optimal fundamental region of the sensor network. We
analyze in detail solutions of the maximum lifetime problem invariant under a
group of isometry transformations of a two dimensional Euclidean plane.
