Abstract
We prove that the automorphism group of a compact 6-manifold M endowed with a symplectic half-flat
{\mathrm{SU}(3)}
-structure has Abelian Lie algebra with dimension bounded by
{\min\{5,b_{1}(M)\}}
. Moreover, we study the properties of the automorphism group action and we discuss relevant examples. In particular, we provide new complete examples on
{T\mathbb{S}^{3}}
which are invariant under a cohomogeneity one action of
{\mathrm{SO}(4)}
.