Non-coding RNAs (ncRNAs)-functional RNA molecules not coding for proteins-are grouped into hundreds of families of homologs. To find new members of an ncRNA gene family in a large genome database, covariance models (CMs) are a useful statistical tool, as they use both sequence and RNA secondary structure information. Unfortunately, CM searches are slow. Previously, we introduced rigorous filters, which provably sacrifice none of CMs accuracy, although often scanning much faster. A rigorous filter, using a profile hidden Markov model (HMM), is built based on the CM, and filters the genome database, eliminating sequences that provably could not be annotated as homologs. The CM is run only on the remainder. Some biologically important ncRNA families could not be scanned efficiently with this technique, largely due to the significance of conserved secondary structure relative to primary sequence in identifying these families. Current heuristic filters are also expected to perform poorly on such families.