Python task: determine if there is a vertical axis of symmetry for a given list of points on the plane.

To determine whether a vertical axis of symmetry exists for a given list of points on a plane, you need to check if a vertical line can be drawn such that all points are symmetric with respect to it.

Solution idea:

1. Find the minimum and maximum x-values among the points.
2. Calculate the potential axis of symmetry as axis = (min_x + max_x) / 2.
3. For each point, check if there exists a point with a mirrored coordinate relative to the axis: if the point is (x, y), then there should be a point (2*axis - x, y).

Python code example:

from collections import defaultdict

def has_vertical_symmetry(points):
    points_set = set(points)
    xs = [x for x, y in points]
    min_x, max_x = min(xs), max(xs)
    axis = (min_x + max_x) / 2
    for x, y in points:
        mirrored = (2*axis - x, y)
        if mirrored not in points_set:
            return False
    return True

# Example usage
points = [(1,1), (3,1), (2,2)]
print(has_vertical_symmetry(points))  # False

points = [(1,1), (3,1), (2,2), (2,2)]
print(has_vertical_symmetry(points))  # True

This method works for a set of points and checks for symmetry relative to a vertical axis.