This is related to an algebraic topology question which uses Van Kampen theorem.
Let $X=A\cup B$ for $A,B$ open and path-connected subspaces of the ambient space.
Suppose that $\pi_1 (A)\cong \mathbb Z^2$ and $\pi_1 (B)\cong 1$.
Also we have that $A\cap B$ is nonempty and path-connected, and $\pi_1 (A\cap B)\cong \mathbb Z$.
Okay, so Van Kampen tell us that \begin{equation*} \pi_1 (X) \cong \pi_1 (A) \ast_{\pi_1 (A\cap B)} \pi_1 (B), \end{equation*} or the amalgamated free product of $\pi_1 (A)$ and $\pi_1(B)$ over $\pi_1 (A\cap B)$.
Switching to the notation of generators, I know that $\mathbb Z^2 = \langle a,b : ab=ba \rangle$ and also that $\mathbb Z = \langle c \rangle$.
Moreover, we will have induced homomorphisms $\phi_\ast : \pi_1 (A\cap B) \hookrightarrow \pi_1 (A)$ and $\psi_\ast : \pi_1 (A\cap B) \hookrightarrow \pi_1 (B)$ and in particular these will be inclusion, i.e., injective maps.
I know that since $B$ is simply connected, therefore $\psi_\ast (\alpha) = 1$ for any $\alpha \in \pi_1 (A\cap B)$, but I don't know what $\phi_\ast (\alpha)$ would be.
I considered $\phi_\ast,$ the inclusion from $\mathbb{Z}$ into $\mathbb Z^2$ but I'm not sure what this would look like, since elements of the latter are ordered pairs whereas $\mathbb Z$ is just single numbers. I believe once I figured out this step then I can get the necessary relations for the group presentation of $\pi_1 (X)$.
As an alternative application of V.K., I know we get $$\pi_1 (X) \cong \left( \pi_1 (A) \ast \pi_1 (B) \right) / N,$$ where $N$ is the smallest normal subgroup of $\pi_1 (A) \ast \pi_1 (B)$ which contains all elements of the form $\phi_\ast (a) \psi_\ast (a)^{-1}$ for all $a\in \pi_1 (A\cap B)$, but once again I run into the issue of figuring out $\phi_\ast (a)$.
My next thought, does it help if I consider breaking up $\pi_1 (A)$ into not $\mathbb Z^2$ but just two individual copies of $\mathbb Z$, and apply V.K. in a different manner?