Could a novae and a white dwarf supernova occur with a white dwarf that is not a member of a binary star system?
No, a white dwarf can't supernova on its own.
A white dwarf is the remains of a dead star. It isn't massive enough or hot enough for any fusion reactions to take place. It is composed of carbon and oxygen.
If a white dwarf is in a close binary configuration with another star. It can accrete material from the other star to increase its mass. Once the mass of the white dwarf approaches the Chandrasekhar limit, which is about 1.44 solar masses, the star collapses under gravity. At this point the temperature rises to the point where carbon fusion can start. A significant amount of the star's mass fuses in a short period triggering a supernova explosion.
So, there is no known mechanism by which a white dwarf can acquire enough material to supernova other than being in a close binary with another star.