  Motivated by numerical evidence of the valence bond groundstate of the
two-dimensional Heisenberg pyrochlore lattice, we argue using a $t$-$J$ model
that it evolves under doping into novel phases characterized by
superconductivity coexisting with the underlying valence-bond solid order. A
fermionic mean-field theory supplemented by exact diagonalization results
provide strong arguments in favor of the stability of such supersolid phases.
The resemblance with modulated superconducting patterns in high-$T_c$ cuprates
as well as possible relevance to frustrated noncuprate superconductors such as
spinels and pyrochlores is discussed.
