On a family of graph manifolds of genus 2

We construct an infinite family of graph manifolds. These manifolds are obtained by gluing a Seifert manifold $(D^2;(2,-1),(2k+1,k))$, where $k\geqslant 1$, and a Seifert manifold $(M^2;(p_1,q_1),(p_2,q_2))$, where $0<q_1,p_1$. The gluing homeomorphism is determined by matrix $\left(
\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}
\right)$ in a natural coordinate systems on boundaries of the Seifert manifolds. We classify those manifolds and prove that all of them have genus two. In addition, for all of them we calculate their first homology groups.