  Biconformal deformations take place in the presence of a conformal foliation,
deforming by different factors tangent to and orthogonal to the foliation.
Four-manifolds endowed with a conformal foliation by surfaces present a natural
context to put into effect this process. We develop the tools to calculate the
transformation of the Ricci curvature under such deformations and apply our
method to construct Einstein $4$-manifolds. One particular family of examples
have ends that collapse asymptotically to ${\mathbb R}^2$.
