Rees algebra and almost linearly presented ideals in three variables

Let $R=\text{k}[x,y,z]$ and $I=(f_0,\dots ,f_{n-1})$ be a height two perfect ideal which is almost linearly presented (that is, all but the last column have linear entries, but the last column has entries which are homogeneous of degree $2$). Further we suppose that after modulo an ideal generated by two variables, the presentation matrix has rank one. Also, the ideal $I$ satisfies $\text{Gs}2$ but not $\text{Gs}3$, then we obtain explicit formulas for the defining ideal of the Rees algebra $\text{rees}(I)$ of $I$.