  Motivated by a search for Lie group structures on groups of Poisson
diffeomorphisms [24], we investigate linearizability of Poisson structures of
Poisson groupoids around the unit section. After extending the Lagrangian
neighbourhood theorem to the setting of cosymplectic Lie algebroids, we
establish that dual integrations of triangular bialgebroids are always
linearizable. Additionally, we show that the (non-dual) integration of a
triangular Lie bialgebroid is linearizable whenever the $r$-matrix is of
so-called cosymplectic type. The proof relies on the integration of a
triangular Lie bialgebroid to a symplectic LA-groupoid, and in the process we
define interesting new examples of double Lie algebroids and LA-groupoids. We
also show that the product Poisson groupoid can only be linearizable when the
Poisson structure on the unit space is regular.
