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We prove the existence of a one-parameter family of nearly parallel G2-structures on the manifold 
$\text{S}^{3}\times \mathbb{R}^{4}$, which are mutually non-isomorphic and invariant under the cohomogeneity one action of the group SU(2)3. This family connects the two locally homogeneous nearly parallel G2-structures that are induced by the homogeneous ones on the sphere S7.
A Riemannian manifold 
$\left( M,\,\rho \right)$ is called Einstein if the metric 
$\rho $ satisfies the condition 
$\text{Ric}\left( \rho \right)\,=\,c\,\cdot \,\rho $ for some constant 
$c$. This paper is devoted to the investigation of 
$G$-invariant Einstein metrics, with additional symmetries, on some homogeneous spaces 
$G/H$ of classical groups. As a consequence, we obtain new invariant Einstein metrics on some Stiefel manifolds 
$SO\left( n \right)/SO\left( l \right)$. Furthermore, we show that for any positive integer 
$p$ there exists a Stiefel manifold $SO\left( n \right)/SO\left( l \right)$ that admits at least $p$
$SO\left( n \right)$-invariant Einstein metrics.
