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ABSTRACT We present adaptive optics-assisted K-band integral field spectroscopy of the central cluster galaxy in 2A 0335+096 ( z= 0.0349). The H2 v=1-0 S(1) emission is concentrated in two peaks within 600 pc of the nucleus and fainter but kinematically active emission extends towards the nucleus. The H2 is in a rotating structure which aligns with, and appears to have been accreted from, a stream of Hα emission extending over 14 kpc towards a companion galaxy. The projected rotation axis aligns with the 5 GHz radio lobes. This H2 traces the known 1.2 × 109 M⊙ CO-emitting reservoir; limits on the Brγ emission confirm that the H2 emission is not excited by star formation, which occurs at a rate of less than 1 M⊙ yr−1 in this gas. If its accretion on to the black hole can be regulated whilst star formation remains suppressed, the reservoir could last for at least 1 Gyr; the simultaneous accretion of just ∼5 per cent of the gas could drive a series of active galactic nucleus (AGN) outbursts which offset X-ray cooling in the cluster core for the full ∼1 Gyr. Alternatively, if the regulation is ineffective and the bulk of the H2 accretes within a few orbital periods (25-100 Myr), the resulting 1062 erg outburst would be among the most powerful cluster AGN outbursts known. In either case, these observations further support cold feedback scenarios for AGN heating. [ABSTRACT FROM AUTHOR]