  Dispersionless bands -- \emph{flatbands} -- provide an excellent testbed for
novel physical phases due to the fine-tuned character of flatband tight-binding
Hamiltonians. The accompanying macroscopic degeneracy makes any perturbation
relevant, no matter how small. For short-range hoppings flatbands support
compact localized states, which allowed to develop systematic flatband
generators in $d=1$ dimension in Phys. Rev. B {\bf 95} 115135 (2017) and Phys.
Rev. B {\bf 99} 125129 (2019). Here we extend this generator approach to $d=2$
dimensions. The \emph{shape} of a compact localized state turns into an
important additional flatband classifier. This allows us to obtain analytical
solutions for classes of $d=2$ flatband networks and to re-classify and
re-obtain known ones, such as the checkerboard, kagome, Lieb and Tasaki
lattices. Our generator can be straightforwardly generalized to three lattice
dimensions as well.
