  We classify isotopy classes of irreducible Heegaard splittings of
solvmanifolds. If the monodromy of the solvmanifold can be expressed as a 2 x 2
matrix with 0 in the lower right hand corner (as always is true when the
absolute value of the trace is 3), then any irreducible splitting is strongly
irreducible and of genus two. If furthermore the absolute value of the trace is
4 or greater, then any two such splittings are isotopic. If the absolute value
of the trace is 3 then, up to isotopy, there are exactly two irreducible
splittings, their associated hyperelliptic involutions commute, and the product
of the involutions is the central involution of the solvmanifold.
  If the monodromy cannot be expressed as a 2 x 2 matrix with 0 in the lower
right hand corner, then the splitting is weakly reducible, of genus three and
unique up to isotopy.
