In this paper, we show that for an Sp(k + 1)-invariant metric ĝ on \(\mathbb{S}^{4k+3}\) (k ⩾ 1) close to the round metric, the conformally compact Einstein (CCE) manifold (M, g) with \((\mathbb{S}^{4k+3},[\hat{g}])\) as its conformai infinity is unique up to isometry. Moreover, by the result in Li et al. (2017), g is the Graham-Lee metric (see Graham and Lee (1991)) on the unit ball B1 ⊂ ℝ4k+4. We also give an a priori estimate of the Einstein metric g. As a by-product of the a priori estimates, based on the estimate and Graham-Lee and Lee’s seminal perturbation results (see Graham and Lee (1991) and Lee (2006)), we directly use the continuity method to obtain an existence result of the non-positively curved CCE metric with prescribed conformal infinity \((\mathbb{S}^{4k+3},[\hat{g}])\) when the metric ĝ is Sp(k + 1)-invariant. We also generalize the results to the case of conformal infinity \((\mathbb{S}^{15},[\hat{g}])\) with ĝ a Spin(9)-invariant metric in the appendix.
