  We investigate the properties of Lindblad equations on $d$-dimensional
lattices supporting a unique steady-state configuration. We consider the case
of a time evolution weakly symmetric under the action of a finite group $G$,
which is also a symmetry group for the lattice structure. We show that in such
case the steady-state belongs to a relevant subspace, and provide an explicit
algorithm for constructing an orthonormal basis of such set. As explicitly
shown for a spin-1/2 system, the dimension of such subspace is extremely
smaller than the dimension of the set of square operators. As a consequence, by
projecting the dynamics within such set, the steady-state configuration can be
determined with a considerably reduced amount of resources. We demonstrate the
validity of our theoretical results by determinining the \emph{exact} structure
of the steady-state configuration of the two dimensional XYZ model in the
presence of uniform dissipation, with and without magnetic fields, up to a
number of sites equal to 12. As far as we know, this is the first time one is
capable of determining the steady-state structure of such model for the 12
sites cluster exactly. Altough in this work we consider explicitly only
spin-1/2 systems, our approach can be exploited in the characterisation of
arbitrary spin systems, fermion and boson systems (with truncated Fock space),
as well as many-particle systems with degrees of freedom having different
statistical properties.
