  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.
