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We present the results of new AGILE observations of PSR B1509-58 performedover a period of $\sim$2.5 years following the detection obtained withpreliminary data. The modulation significance of the lightcurve above 30 MeV isat a 5$\sigma$ confidence level and the lightcurve is similar to those foundearlier up to 30 MeV by COMPTEL: a broad asymmetric first peak reaching itsmaximum $0.39 \pm 0.02$ cycles after the radio peak plus a second peak at $0.94\pm 0.03$. The $\gamma$-ray spectral energy distribution of pulsed flux is welldescribed by a power-law (photon index $\alpha=1.87\pm0.09$) with a remarkablecutoff at $E_c=81\pm 20$ MeV, representing the softest spectrum observed among$\gamma$-ray pulsars so far. The unusual soft break in the spectrum of PSRB1509-58 has been interpreted in the framework of polar cap models as asignature of the exotic photon splitting process in the strong magnetic fieldof this pulsar. In the case of an outer-gap scenario, or the two pole causticmodel, better constraints on the geometry of the emission would be needed fromthe radio band in order to establish whether the conditions required by themodels to reproduce AGILE lightcurves and spectra match the polarizationmeasurements.

AGILE observations of PSR B1509-58: is QED photon splitting at work in pulsars?