Summary: The classical boundary-value problem of the Einstein field equations is studied with an arbitrary cosmological constant, in the case of a compact \((S^3)\) boundary given a biaxial Bianchi-IX positive-definite three-metric, specified by two radii \((a,b)\). For the simplest, four-ball topology of the manifold with this boundary, the regular classical solutions are found within the family of Taub-NUT-(anti) de Sitter metrics with self-dual Weyl curvature. For arbitrary choice of positive radii \((a,b)\), we find that there are three solutions for the infilling geometry of this type. We obtain exact solutions for them and for their Euclidean actions. The case of negative cosmological constant is investigated further. For reasonable squashing of the three-sphere, all three infilling solutions have real-valued actions which possess a “cusp catastrophe” structure with a non-self-intersecting “catastrophe manifold” implying that the dominant contribution comes from the unique real positive-definite solution on the ball. The positive-definite solution exists even for larger deformations of the three-sphere, as long as a certain inequality between a and b holds. The action of this solution is proportional to \(-a^3\) for large a \((\sim b)\) and hence larger radii are favoured. The same boundary-value problem with more complicated interior topology containing a “bolt” is investigated in a forthcoming paper.