The structure of Sally modules of rank one

A complete structure theorem of Sally modules of $\text{fkm}$-primary ideals $I$ in a Cohen-Macaulay local ring $(A,\text{m})$ satisfying the equality ${\text{e}}_1(I)={\text{e}}_0(I)-{\mathrm{\ell }}_A(A\mathrm{/}I)+1$ is given, where ${\text{e}}_0(I)$ and ${\text{e}}_1(I)$ denote the first two Hilbert coefficients of $I$.