Twistor space of a generalized quaternionic manifold

We first make a little survey of the twistor theory for hypercomplex, generalized quaternionic or manifolds. This last was initiated by Pantilie (Ann. Mat. Pura. Appl. 193 (2014) 633–641), and allows one to extend Penrose correspondence from quaternion case. He showed that any almost manifold equipped with an appropriate connection admit space which comes naturally tautological complex structure. But he has left open problem integrability. The aim this article is give integrability criterion structure some examples especially in case hyperkähler manifolds using Bismut connection, introduced Gualtieri (Branes on Poisson varieties, many facets geometry: tribute Nigel Hitchin (2010) (Oxford: Oxford University Press) pp. 368–395).