Counting integral points on indefinite ternary quadratic equations over number fields

Abstract We study an asymptotic formula for counting integral points over equation defined by a non-degenerate indefinite ternary quadratic form f representing non-zero integer such that $-a\cdot det(\,f)$ is square number field. In particular, we prove the leading coefficient of this given product local densities normalized $1-p^{-1}$ all finite primes p.