Summary: We find instanton/cosmological solutions with biaxial Bianchi-IX symmetry, involving nontrivial spatial dependence of the \(\mathbb CP^1\)- and \(\mathbb CP^2\)-sigma-models coupled to gravity. Such manifolds arise in \(N = 1\), \(d = 4\) supergravity with supermatter actions and hence the solutions can be embedded in supergravity. There is a natural way in which the standard coordinates of these manifolds can be mapped into the four-dimensional physical space. Due to its special symmetry, we start with \(\mathbb CP^2\) with its corresponding scalar ansatz; this further requires the spacetime to be \(SU(2) \times U(1)\)-invariant. The problem then reduces to a set of ordinary differential equations whose analytical properties and solutions are discussed. Among the solutions there is a surprising, special family of exact solutions which owe their existence to the nontrivial topology of \(\mathbb CP^2\) and are in 1-1 correspondence with matter-free Bianchi-IX metrics. These solutions can also be found by coupling \(\mathbb CP^1\) to gravity. The regularity of these Euclidean solutions is discussed-the only possibility is bolt-type regularity. The Lorentzian solutions with similar scalar ansatz are all obtainable from the Euclidean solutions by Wick rotation.