Let $\mathbb{C} \subset R \subset \mathbb{C}[x,y]$ be a noetherian integral domain.
Further assume that: (1) $\mathbb{C}[x,y]$ is separable over $R$. (2) $\mathbb{C}[x,y]$ is algebraic over $R$ (integral, if this helps). (3) The field of fractions of $R$ is $\mathbb{C}(x,y)$.
Do NOT assume that: (4) $R$ is integrally closed. (5) $\mathbb{C}[x,y]$ is a flat $R$-module.
My question: What additional conditions, which are not one of (4) or (5), are needed in order to guarantee the normality of $R$ or the flatness of $R \subset \mathbb{C}[x,y]$?
(In a more specific $R$ I have in mind, I can show that normality of $R$ is equivalent to flatness of $R \subset \mathbb{C}[x,y]$).
Any help will be appreciated.
EDIT: (i) Notice that I do not require to know when such $R$ is regular; normal is enough. (ii) I am familiar with Serre's criterion for normality, but I do not know how it may help me in my specific case.