  McCallum-style Cylindrical Algebra Decomposition (CAD) is a major improvement
on the original Collins version, and has had many subsequent advances, notably
for total or partial equational constraints. But it suffers from a problem with
nullification. The recently-justified Lazard-style CAD does not have this
problem. However, transporting the equational constraints work to Lazard-style
does reintroduce nullification issues. This paper explains the problem, and the
solutions to it, based on the second author's Ph.D. thesis and the
Brown--McCallum improvement to Lazard.
  With a single equational constraint, we can gain the same improvements in
Lazard-style as in McCallum-style CAD . Moreover, our approach does not fail
where McCallum would due to nullification. Unsurprisingly, it does not achieve
the same level of improvement as it does in the non-nullified cases. We also
consider the case of multiple equational constraints.
