On an inverse problem for restricted sumsets

Let $n$ be a positive integer, and let $A$ be a set of $k\ge 2n-1$ integers. For the restricted sumset $$S_n(A)=\{a_1+\cdots +a_n:a_1,\dots ,a_n\in A,\text{ and }{a}_i^2\mathrm{\ne }{a}_j^2\text{ for }1\le i<j\le n\},$$ by a 2002 result of Liu and Sun we have $$\mathrm{\mid }{S}_n(A)\mathrm{\mid }\ge (k-1)n-\frac{3}{2}n(n-1)+1.$$ In this paper, we determine the structure of $A$ when the lower bound is attained.