  The tachyon vacuum solution of Schnabl is based on the wedge states, which
close under the star product and interpolate between the identity state and the
sliver projector. We use reparameterizations to solve the long-standing problem
of finding an analogous family of states for arbitrary projectors and to
construct analytic solutions based on them. The solutions simplify for special
projectors and allow explicit calculations in the level expansion. We test the
solutions in detail for a one-parameter family of special projectors that
includes the sliver and the butterfly. Reparameterizations further allow a
one-parameter deformation of the solution for a given projector, and in a
certain limit the solution takes the form of an operator insertion on the
projector. We discuss implications of our work for vacuum string field theory.
