Constraints on Energy Independent Solutions of the Solar Neutrino Problem

We analyze the latest published solar neutrino data assuming an arbitrary neutrino oscillation/conversion mechanism suppresses the electron neutrino flux from the Sun independent of energy. For oscillations/transitions into active (sterile) neutrinos such mechanisms are ruled out at 99.96 (99.9997) % C.L. assuming the standard solar model by Bahcall and Pinnsoneault '95 correctly predicts all solar neutrino fluxes within their estimated uncertainties. Even if one allows for $^8{\rm B}$ and $^7{\rm Be}$ solar neutrino fluxes that are vastly different from the ones in contemporary standard solar models these mechanisms are strongly disfavored by the data.