We consider a two-band model with strong correlations derived from the Periodic Anderson Model by means of first order perturbation theory with respect to 1/U (U is the local interaction for f-sites). We show that at half-filling the Gutzwiller ansatz is an exact solution of that model for special parameters, generalizing a result of Brand and Giesekus. We use an exponential representation of the wave function, which allows for the derivations to use a terminating multi-commutator expansion. The latter motivates a generaliztion of the Gutzwiller wave function for the half-filled case. The combination of the Bardeen Cooper Schrieffer wave function and the Gutzwiller ansatz leads to exact solutions for further two-band models with strong correlations.