The number of representations of squares by integral ternary quadratic forms

Let f be a positive definite integral ternary quadratic form and let r(k,f) be the number of representations of an integer k by f. In this article we study the number of representations of squares by f. We say the genus of f, denoted by gen(f), is indistinguishable by squares if for every integer n, r(n2,f)=r(n2,f′) for every quadratic form f′∈gen(f). We find some non-trivial genera of ternary quadratic forms which are indistinguishable by squares. We also give some relation between indistinguishable genera by squares and the conjecture given by Cooper and Lam, and we resolve their conjecture completely.