Let's take for example $x^2 + y^2 = 1.$ This has infinitely many symmetries, but a portion of those symmetries have the form $(x,y) \rightarrow (x \cos(\epsilon) - y \sin( \epsilon), x \sin( \epsilon) + y \cos( \epsilon))$ for $\epsilon \in [-\pi, \pi].$
My questions are: how do you derive such symmetries for algebraic equations and systems of algebraic equations, and, do these symmetries relate to their solutions in any way?