Abstract
In this article we provide a description of the local structure of closed symmetric 2-differentials on a complex surface. The main technical result of the article is that a sum of two local singular analytic functions which both are constant along two different smooth holomorphic foliations can be locally holomorphic only if both functions have at most meromorphic singularities. As a corollary we proof that locally a product of two singular closed differentials on a surface is holomorphic only if the singularities of the differentials are at most exponents of local meromorphic functions.