The structure of a solvmanifold's Heegaard splittings
  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. 
Daryl Cooper and Martin Scharlemann