They discover an obelisk with strange markings, and observe a settlement whose inhabitants, according to Spock, are descended from Native Americans (Mohican, Navaho and Delaware).
Spock and McCoy are unable to locate Kirk and return to the Federation starship USS Enterprise to complete their mission.
In the meantime Spock, taking command of the Enterprise, drives her at maximum warp speed to beat the approaching asteroid to the deflection point.
The ship's deflectors result in a change of only 0.0013 degrees in its course, which Mr. Sulu tells Spock is insufficient to avert collision with the planet.
Having exhausted their options, the Enterprise heads back to the planet on impulse power to attempt to recover Captain Kirk and devise another solution to the asteroid problem.
It is an asteroid deflector built by the "Preservers", an ancient race that resettled various endangered humanoid populations on other planets in order to ensure their survival.
[1] Author Jon Davidson on his sci-fi tv and movie website rated this episode 7 out of 10, noting "while the presence of Native Americans on yet another parallel Earth planet is only briefly explained, The Paradise Syndrome provides a compelling romance subplot to compensate for any questionable science fiction aspects.
[6] This episode was released in Japan on December 21, 1993 as part of the complete season 3 LaserDisc set, Star Trek: Original Series log.3.