  We have obtained deep narrowband images in the near-infrared H2 (2.122
microns) emission line for a sample of 15 faint Isaac Newton Telescope
Photometric Halpha Survey (IPHAS) bipolar planetary nebulae (PNe) to search for
molecular material. H2 emission is found in most of them (14 out of 15), mostly
associated with rings at their equatorial regions and with their bipolar lobes.
These detections add to the high occurrence of H2 emission among bipolar PNe
reported in previous works, resulting from the large reservoir of molecular
material in these sources and the suitable excitation conditions for H2
emission. The correlation between detailed bipolar morphology and H2 luminosity
is also confirmed: bipolar PNe with broad equatorial rings (R-BPNe) have almost
no continuum emission, are H2 brighter and have larger H2/Brgamma line ratio
than bipolar PNe with pinched equatorial waists (W-BPNe). The origin of this
dichotomy is unclear. The larger size and age of R-BPNe are consistent with
shock excitation of H2, whereas ultraviolet pumping is most likely the
excitation mechanism in the smaller and younger W-BPNe, which would explain
their lower H2 luminosity. Although both types of bipolar PNe seem to proceed
from the same progenitor population, this does not imply that R-BPNe descend
from W-BPNe. Otherwise, we note that some of the H2-weak bipolar PNe harbor
post-common envelope binary systems and symbiotic stars. Finally, we suggest
that the long-living H2 emission from R-BPNe arises from a discrete
distribution of compact knots embedded within the ionized gas at the equatorial
region.
