In this paper, we study asymptotic expansions of positive solutions of the conformal scalar curvature equation \( - \Delta u = K(x) u^\frac{n + 2}{n - 2} ~~~~~~ \text {in} ~ B_1 \setminus \{ 0 \} \) with an isolated singularity at the origin. Under certain flatness conditions on K, we establish a higher-order expansion of solutions near the origin. In particular, we give the exact second-order asymptotic expansion of solutions when \(n \ge 6\) . Moreover, we also obtain an arbitrary-order expansion of singular positive solutions of the anisotropic elliptic equation \( - \,\textrm{div} (|x|^{- 2 a} \nabla u) = |x|^{- b p} u^{p - 1} ~~~~~~ \text {in} ~ B_1 \setminus \{ 0 \}, \) where \(0 \le a < \frac{n - 2}{2}\) , \(a \le b < a + 1\) and \(p = \frac{2 n}{n - 2 + 2 (b - a)}\) . This equation arises from the celebrated Caffarelli-Kohn-Nirenberg inequality.
