Circle compactifications of Minkowski$_D$ solutions, flux vacua and solitonic branes

G-structure techniques are used to construct broad classes of circle compactifications of Mink$_{D+1}$ solutions to Mink$_{D}$ embedded into type II supergravity for $D=1,...5$. Under a certain assumptions we show that the conditions that imply supersymmetry for Mink$_{D+1}$ imply those of the Mink$_{D}$ solution, but that Bianchi identities of the fluxes must be modified. This realises an off shell solution generating technique for supersymmetric solutions or a "supersymmetry generating" technique. Along the way it is necessary for us to derive G structure conditions for general ${\cal N}=(1,0)$ supersymmetric Mink$_2$ solutions and a restricted class of Mink$_1$ solutions. We apply our results to construct some simple Minkowski flux vacua before turning our attention to "solitonic branes" which are generalisations of the AdS soliton. We are able to generalise known examples in two ways: 1) to embed them in terms of generic Sasaki Einstein manifolds. 2) To modify the harmonic factor to include D$p$ brane sources at one end of the space.