  We study certain subspaces of solutions to the sl_2 rational qKZ equation at
level zero. Each subspace is specified by the vanishing of the residue at a
certain divisor which stems from models in two dimensional integrable field
theories. We determine the character of the subspace which is parametrized by
the number of variables and the sl_2 weight of the solutions. The sum of all
characters with a fixed weight gives rise to the branching functions of the
irreducible representations of sl_2 in the level one integrable highest weight
representations of \hat{sl_2}. It is written in the fermionic form.
