About the geometry of almost para-quaternionic manifolds

We provide a general criterion for the integrability of the almost para-quaternionic structure of an almost para-quaternionic manifold (M,P)(M,P) of dimension 4m⩾84m⩾8 in terms of the integrability of two or three sections of the defining rank three vector bundle PP. We relate it with the integrability of the canonical almost complex structure of the twistor space and with the integrability of the canonical almost para-complex structure of the reflector space of (M,P)(M,P). We deduce that (M,P)(M,P) has plenty of locally defined, compatible, complex and para-complex structures, provided that PP is integrable.