AXP's are
believed to be spinning down pulsars, apparently not powered by
accretion from a companion star, and with a luminosity larger than the
available rotational energy loss of a neutron star. The two main
models are magnetars (highly magnetized - 1014 G - neutron stars)
or accretion from a residual disc. 4U0142+61 was the first AXP to be
optically identified.
Hulleman, van Kerkwijk, Kulkarni, 2000, Nature, 408, 689