  We introduce an $\mathfrak{F}$-valued generalization of the Virasoro algebra,
called the Frobenius-Virasoro algebra $\mathfrak{vir_F}$, where $\mathfrak{F}$
is a Frobenius algebra over $\mathbb{R}$. We also study Euler equations on the
regular dual of $\mathfrak{vir_F}$, including the $\mathfrak{F}$-$\mathrm{KdV}$
equation and the $\mathfrak{F}$-$\mathrm{CH}$ equation and the
$\mathfrak{F}$-$\mathrm{HS}$ equation, and discuss their Hamiltonian
properties.
