Positive Einstein metrics with as the principal orbit

We prove that there exists at least one positive Einstein metric on HPᵐ⁺¹ HP̄ᵐ⁺¹ for m≥ ~2 . Based on the existence of the first Einstein metric, we give a criterion to check the existence of a second Einstein metric on HPᵐ⁺¹ HP̄ᵐ⁺¹ . We also investigate the existence of cohomogeneity-one positive Einstein metrics on S⁴ᵐ⁺⁴ and prove the existence of a non-standard Einstein metric on S⁸ .