The Explored Galaxy: The location of Beta Geminorum and Pollux IV
The Beta Geminorum system, or Pollux system, was a star system with at least five planets. In the 23rd century, there was a strange lack of indigenous intelligent life in the system, when it was explored by the USS Enterprise in 2267. The location of Beta Geminorum was depicted on a Federation star chart in 2293. (Star Trek VI: The Undiscovered Country)
Pollux IV was the ancient homeworld of the Greek gods. (TOS: "Who Mourns for Adonais?")
Background
Michael Okuda, who created the chart The Explored Galaxy, believed that Beta Geminorum and Pollux were two distinct stars. So, he created two systems, the Beta Geminorum and the Pollux, and located them in the same region of space. Unlike the situation with another system, Canopus, where the canon never connected Canopus to Alpha Carinae, the canon did established that connection with Beta Geminorum and Pollux. There might be reasons that could be postulated to explain this situation, but without a statement from Mr. Okuda, the definitive reason for this retcon can't be known.
The Explored Galaxy
This chart was also seen in several Star Trek: The Next Generation and Star Trek: Deep Space Nine episodes set in the 24th century, from the year 2364 to 2370. These were: TNG: "Conspiracy", "The Measure Of A Man", "The Emissary", "The Mind's Eye", "The Game" and DS9: "In the Hands of the Prophets", "Cardassians".
Star Trek: Star Charts
Unlike Mr. Okuda, Geoffrey Mandel believed that Beta Geminorum and Pollux were the same system. This system was located in the Beta Quadrant. System's primary was a K-class star. This star had a magnitude of -1, which was 100 times brighter than Sol. (page 60)
Apocrypha
Pollux V was also featured in the video game Star Trek: 25th Anniversary.