  (Abridged) We studied an exceptional period of activity of the anomalous
X-ray pulsar 1E 1547.0-5408 in January 2009, during which about 200 bursts were
detected by INTEGRAL. The major activity episode happened when the source was
outside the field of view of all the INTEGRAL instruments. But we were still
able to study the properties of 84 bursts detected simultaneously by the
anti-coincidence shield of the spectrometer SPI and by the detector of the
imager ISGRI. We find that the luminosity of the 22 January 2009 bursts of 1E
1547.0-5408 was > 1e42 erg/s. This luminosity is comparable to that of the
bursts of soft gamma repeaters (SGR) and is at least two orders of magnitude
larger than the luminosity of the previously reported bursts from AXPs.
Similarly to the SGR bursts, the brightest bursts of 1E 1547.0-5408 consist of
a short spike of ~100 ms duration with a hard spectrum, followed by a softer
extended tail of 1-10 s duration, which occasionally exhibits pulsations with
the source spin period of ~2 s. The observation of AXP bursts with luminosities
comparable to the one of SGR bursts strengthens the conjecture that AXPs and
SGRs are different representatives of one and the same source type.
