Special solutions to the Type IIA flow

We consider the source-free Type IIA flow introduced by Fei–Phong–Picard–Zhang $\href{https://dx.doi.org/10.4310/CJM.2021.v9.n3.a3}{\textrm{[10]}}$, and we study it in the case where the relevant geometric datum is a symplectic half-flat SU(3)-structure. We show the existence of ancient, immortal and eternal solutions to the flow, provided that the initial symplectic half-flat structure satisfies suitable properties. In particular, we prove that the solution starting at a symplectic half-flat structure with Hermitian Ricci tensor is ancient and evolves self-similarly by scaling the initial datum. These results apply to all known (locally) homogeneous spaces admitting invariant symplectic half-flat SU(3)-structures.