Bianchi type A hyper-symplectic and hyper-K\"ahler metrics in 4D
  We present a simple explicit construction of hyper-Kaehler and hyper-symplectic (also known as neutral hyper-Kaehler or hyper-parakaehler) metrics in 4D using the Bianchi type groups of class A. The construction underlies a correspondence between hyper-Kaehler and hyper-symplectic structures in dimension four. 
Luis C. de Andr\'es, Marisa Fern\'andez, Stefan Ivanov, Jos\'e A.   Santisteban, Luis Ugarte, Dimiter Vassilev
math.DG hep-th