Are there any solar systems without a sun? 1) Are we able to study any other solar systems, that once contained a sun the size of our own... that has since died, or gone Red Giant, the way it is predicted our sun will die?
2) If we have been able to study such systems, can we tell if their planets still own or still generate, a magnetosphere... In the absence of a living star in their system? How dependent is a planet on a star for it's magnetosphere?
 A: For your first question, a requirement for a solar system is that there must be at least one star contained within it. Without a star or some other intense gravity field holding the planets in orbit, the planets would drift away. It is possible for any class of star- from dwarf to supergiant- to hold planets in orbit and therefore have a solar system. 
Current technology cannot identify magnetospheres around exoplanets due to our limited processes. However it is reasonable to assume that a planet in the habitable zone of a star may possess a magnetosphere. 
A planet's magnetosphere is dependent not on a star, but on the rotation of that planet's core. Earth's core rotates within the mantle, creating our supposedly unique magnetosphere. The only real interaction between a star and a planet's magnetosphere is the magnetosphere deflecting solar winds from the star.
A: There is plenty of evidence for planets around evolved stars (giants) and there is also plenty of evidence for planetary material around white dwarfs and planets around neutron stars.
Planets around red giants are primarily found using the Doppler technique (the planets are too small compared with the star to produce a significant transit signal). See for example (there are many others) Lee et al. (2012).
Planetary material around white dwarfs is inferred from chemical pollution of their atmospheres with heavy elements and metals (e.g. Farihi et al. 2010). Since such material should sink quickly, it must be being replenished. Planets have not yet been found - a transit in front of a tiny white dwarf would be rare, and it is difficult to measure their velocities precisely enough to use Doppler methods.
Finally, the first ever exoplanets were discovered around a neutron star by Wolszcznan & Frail (1992). These were discovered using Doppler shift in the neutron star radio pulsation frequency. So clearly at least some planets can survive the evolution and death of their stars.
A: I'm not sure if there are any planetary systems without a star. But in answer to your question:  No, because a solar system is so called because there is a sun -hence the name solar which is of a sun. I don't believe we could currently detect planets without a sun as we first detect the star then measure the orbits of its planetary bodies. Hope this helps - you could most likely get a more satisfactory answer by contacting NASA or a astrophysics department of a university.
A: You bet.
There are a number of pulsar planets known, which are, as the name implies, worlds orbiting pulsars. The nearest confirmed system is PSR B1257+12, at a distance of 22 000 Pm (2300 ly) from Earth and it actually hosts three planets. Granted, a pulsar is formed from a star larger than the one you mentioned which was the size of the Sun, but it's still effectively the same thing: a planetary system whose central body is no longer an active star. The three planets actually have names: Draugr, Poltergeist, and Phobetor. Draugr, the closest planet, is on the order of magnitude of the mass of our Moon and orbits the central pulsar in about 2.2 megaseconds (25 days).
Sources:
http://www.openexoplanetcatalogue.com/planet/PSR%201257+12%20B/
https://en.wikipedia.org/wiki/PSR_B1257%2B12
