Normal extensions of linear operators

Let $L_0$ be a densely defined minimal linear operator in a Hilbert space $H$. We prove that if there exists at least one correct extension $L_S$ of $L_0$ with the property $D(L_S ) = D(L^*_S )$, then we can describe all correct extensions $L$ with the property $D(L) = D(L^*)$. We also prove that if $L_0$ is formally normal and there exists at least one correct normal extension $L_N$, then we can describe all correct normal extensions $L$ of $L_0$. As an example, the Cauchy–Riemann operator is considered.