I am looking for an algorithm that can be used on any equation that contains polynomials containing x and y to determine if reflective or rotational symmetries exist. If it is possible, I would like a method that works for all algebraic equations to make it slightly more general. I can graph any function to quickly look at the graph for a visual confirmation of symmetry but that is not precise and it does not tell me exactly if the symmetry I see is on the y axis or very close epsilon to the y axis where visual methods fail. Another concern would be that I can not use the visual obviously method on an algebraic proof that symmetry does exist. What book and what chapter should I start looking for the area of mathematics that is concerned with such mathematical methods.
I did not find this on stack exchange with a basic search.