Could a novae and a white dwarf supernova occur with a white dwarf that is not a member of a binary star system?

The remnants of a dead star are called white dwarfs, and they are made of carbon and oxygen. They are not massive or hot enough for any fusion reactions to occur.

The star collapses under gravity when the mass of the white dwarf approaches the Chandrasekhar limit, or 1.44 solar masses. At this point, the temperature rises to the point where carbon fusion can start, and a sizable portion of the star's mass fuses in a short amount of time, triggering a supernova explosion. 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.

Hence, other than being in a close binary with another star, there is no known way for a white dwarf to gather enough material to supernova.

No, a white dwarf must be in a binary star system in order for either novae or white dwarf supernovae to occur. A white dwarf can only undergo novae if material from a companion star accretes onto its surface and causes a thermonuclear explosion. A white dwarf can only undergo white dwarf supernovae if it accretes enough mass from its companion star to reach the Chandrasekhar limit and causes a catastrophic explosion.