We present a systematic single-dish search for molecular outflows toward a sample of 9 candidate low-luminosity protostars and 30 candidate Very Low Luminosity Objects (VeLLOs; L_int < 0.1 L_sun). The sources are identified using data from the Spitzer Space Telescope catalogued by Dunham et al. toward nearby (D < 400 pc) star forming regions. Each object was observed in 12CO and 13CO J = 2-1 simultaneously using the sideband separating ALMA Band-6 prototype receiver on the Heinrich Hertz Telescope at 30 arcsecond resolution. Using 5-point grid maps we identify five new potential outflow candidates and make on-the-fly maps of the regions surrounding sources in the dense cores B59, L1148, L1228, and L1165. Of these new outflow candidates, only the map of B59 shows a candidate blue outflow lobe associated with a source in our survey. We also present larger and more sensitive maps of the previously detected L673-7 and the L1251-A IRS4 outflows and analyze their properties in comparison to other outflows from VeLLOs. The accretion luminosities derived from the outflow properties of the VeLLOs with detected CO outflows are higher than the observed internal luminosity of the protostars, indicating that these sources likely had higher accretion rates in the past. The known L1251-A IRS3 outflow is detected but not remapped. We do not detect clear, unconfused signatures of red and blue molecular wings toward the other 31 sources in the survey indicating that large-scale, distinct outflows are rare toward this sample of candidate protostars. Several potential outflows are confused with kinematic structure in the surrounding core and cloud. Interferometric imaging is needed to disentangle large-scale molecular cloud kinematics from these potentially weak protostellar outflows.
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