  We study a family of Fredholm determinants associated to deformations of the
sine kernel, parametrized by a weight function w. For a specific choice of w,
this kernel describes bulk statistics of finite temperature free fermions. We
establish a connection between these determinants and a system of
integro-differential equations generalizing the fifth Painlev\'e equation, and
we show that they allow us to solve an integrable PDE explicitly for a large
class of initial data.
