  In the authors's previous work on symmetric differentials and their
connection to the topological properties of the ambient manifold, a class of
symmetric differentials was introduced: closed symmetric differentials
([BoDeO11] and [BoDeO13]). In this article we give a description of the local
structure of closed symmetric 2-differentials on complex surfaces, with an
emphasis towards the local decompositions as products of 1-differentials. We
show that a closed symmetric 2-differential $w$ of rank 2 (i.e. defines two
distinct foliations at the general point) has a subvariety $B_w\subset X$
outside of which $w$ is locally the product of closed holomorphic
1-differentials. The main result, theorem 2.6, gives a complete description of
a (locally split) closed symmetric 2-differential in a neighborhood of a
general point of $B_w$. A key feature of theorem 2.6 is that closed symmetric
2-differentials still have a decomposition as a product of 2 closed
1-differentials (in a generalized sense) even at the points of $B_w$. The
(possibly multi-valued) closed 1-differentials can have essential singularities
along $B_w$, but one still has a control on these essential singularities. The
essential singularities come from exponentials of meromorphic functions
acquiring poles along the irreducible components of $B_w$ of order bounded by
the order of contact of the 2 foliations defined by the symmetric
2-differential along that irreducible component.
