Iwasawa module of the cyclotomic ${\mathbb{Z}}_2$-extension of certain real quadratic fields

For a real quadratic field $K=\mathbb{Q}(\sqrt{D})$, let $K_{\mathrm{\infty }}$ denote the cyclotomic ${\mathbb{Z}}_p$-extension of $K$. Greenberg conjectured that the corresponding Iwasawa module $X_{\mathrm{\infty }}$ is finite. Building on the work of Mouhib and Movahhedi, we provide new examples of real quadratic fields for which the conjecture holds, when $X_{\mathrm{\infty }}$ is cyclic and the prime is $p=2$. Furthermore, we find a fundamental system of units for certain biquadratic fields of the form $\mathbb{Q}(\sqrt{2},\sqrt{D})$ and show how to use it to calculate the order of $X_{\mathrm{\infty }}$.