  A geometric description of the first Poisson cohomology groups is given in
the semilocal context, around (possibly singular) symplectic leaves. This
result is based on the splitting theorems for infinitesimal automorphisms of
coupling Poisson structures which describe the interaction between the
tangential and transversal data of the characteristic distributions. As a
consequence, we derive some criteria of vanishing of the first Poisson
cohomology groups and apply the general splitting formulas to some particular
classes of Poisson structures associated with singular symplectic foliations.
