  The normalized eigenvalues $\Lambda_i(M,g)$ 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 the present paper a family of
extremal metrics induced by minimal immersions in $\mathbb{S}^5$ is provided.
