  When is the collection of $\mathsf S$-Toeplitz operators with respect to a
tuple of commuting bounded operators $\mathsf S= (S_1, S_2, \ldots , S_{d-1},
P)$, which has the symmetrized polydisc as a spectral set, non-trivial? The
answer is in terms of powers of $P$ as well as in terms of a unitary extension.
En route, Brown-Halmos relations are investigated. A commutant lifting theorem
is established. Finally, we establish a general result connecting the
$C^*$-algebra generated by the commutant of $\mathsf S$ and the commutant of
its unitary extension $\mathsf R$.
