The symbols of the generating Toeplitz operators are chosen to be suitable extensions to 882a8ed90c134c8d437fec2b5699" title="Click to view the MathML source">B2 of families 872ac9c465fdf222e9c10b" title="Click to view the MathML source">S of bounded functions on e9fbd" title="Click to view the MathML source">D. Symbol classes 872ac9c465fdf222e9c10b" title="Click to view the MathML source">S that generate important classical commutative and non-commutative Toeplitz algebras in 870e2dc47bd1f5"> are of particular interest. In this paper we discuss various examples. In the case of e5f13752e05d1a391135e630db690b9a"> and 887a"> we characterize all irreducible representations of the resulting Toeplitz operator C⁎-algebras. Their Calkin algebras are described and index formulas are provided.