  Poisson transversals are those submanifolds in a Poisson manifold which
intersect all symplectic leaves transversally and symplectically. In a previous
note we proved a normal form theorem around such submanifolds. In this
communication, we promote that result to a normal form theorem for Poisson maps
around Poisson transversals. A Poisson map pulls a Poisson transversal back to
a Poisson transversal, and our first main result states that simultaneous
normal forms exist around such transversals, for which the Poisson map becomes
transversally linear, and intertwines the normal form data of the transversals.
  Our second main result concerns symplectic integrations. We prove that a
neighborhood of a Poisson transversal is integrable exactly when the Poisson
transversal itself is integrable, and in that case we prove a normal form
theorem for the symplectic groupoid around its restriction to the Poisson
transversal, which puts all its structure maps in normal form.
  We conclude the paper by illustrating our results with examples arising from
Lie algebras.
