For a compact Lie group, we ask whether an equivariant symplectic k-blowup of the projective plane or a ruled surface is determined up to equivariant isotopy by the sizes of the blowups and the fixed components at which they are performed. We focus on Hamiltonian circle actions and homologically trivial cyclic actions. In the non-equivariant setting, this question was answered by McDuff, applying inflation to obtain an isotopy out of a deformation of cohomologous symplectic forms. However, McDuff's key step of applying Taubes' "Seiberg-Witten equals Gromov" to find J-holomorphic curves in the required classes does not hold in the equivariant setting: it does not guarantee the existence of J-holomorphic curves for some invariant tamed almost complex structure. We bypass this step by explicitly finding J-holomorphic curves for an invariant tamed almost complex structure in certain classes, e.g., by reading them from the decorated graph associated to a Hamiltonian circle action.