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The normalized eigenvalues 
${{\Lambda }_{i}}\left( M,\,g \right)$ of the Laplace–Beltrami operator can be considered as functionals on the space of all Riemannian metrics 
$g$ on a fixed surface 
$M$. In recent papers several explicit examples of extremal metrics were provided. These metrics are induced by minimal immersions of surfaces in 
${{\mathbb{S}}^{3}}$ or 
${{\mathbb{S}}^{4}}$. In this paper a family of extremal metrics induced by minimal immersions in 
${{\mathbb{S}}^{5}}$ is investigated.
