  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. 
Algebraic Surfaces with $p_g=q=1, K^2=4$ and Genus 3 Albanese Fibration