On balanced HKT manifolds

We prove the openness of the balanced HKT cone within the cone of HKT structures on a compact hypercomplex manifold (M, I, J, K). We also study the Lie algebra of hyperholomorphic vector fields of type (1, 0) with respect to I, with particular emphasis on the case when there exists a compatible balanced HKT metric. These fields exhibit a strict interplay with the balanced HKT structure, for instance, we prove a harmonicity property for (1, 0) -forms dual to hyperholomorphic vector fields. We also show non-existence of hyperholomorphic (1, 0) -vector fields on some hypercomplex manifolds admitting a HKT--Einstein metric.