  We consider exceptional vertex operator algebras and vertex operator
superalgebras with the property that particular Casimir vectors constructed
from the primary vectors of lowest conformal weight are Virasoro descendents of
the vacuum. We show that the genus one partition function and characters for
simple ordinary modules must satisfy modular linear differential equations. We
show the rationality of the central charge and module lowest weights,
modularity of solutions, the dimension of each graded space is a rational
function of the central charge and that the lowest weight primaries generate
the algebra. We also discuss conditions on the reducibility of the lowest
weight primary vectors as a module for the automorphism group. Finally we
analyse solutions for exceptional vertex operator algebras with primary vectors
of lowest weight up to 9 and for vertex operator superalgebras with primary
vectors of lowest weight up to 17/2. Most solutions can be identified with
simple ordinary modules for known algebras but there are also four conjectured
algebras generated by weight two primaries and three conjectured extremal
vertex operator algebras generated by primaries of weight 3, 4 and 6
respectively.
