  In this paper, we study the Gieseker moduli space $\mathcal{M}_{1,1}^{4,3}$
of minimal surfaces with $p_g=q=1, K^2=4$ and genus 3 Albanese fibration. Under
the assumption that direct image of the canonical sheaf under the Albanese map
is decomposable, we find two irreducible components of
$\mathcal{M}_{1,1}^{4,3}$, one of dimension 5 and the other of dimension 4.
