(1) Because 4 > 23, the trajectory is an ellipse with F 1 and F2 as the focus.

Taking the midpoint of the line segment F 1F2 as the coordinate origin and the straight line F 1F2 as the x axis, a plane rectangular coordinate system is established.

The γ equation of the trajectory of the moving point A is x24+y2 =1;

(2)① From the meaning of the question, |F 1F2|=23. When a is at the intersection of the ellipse and the Y axis, △AF 1F2 has the largest height and area.

∴△△AF1F2 The maximum area is 12? 23? 1=3;

② When AC ⊥ F1F2, besides A and C, there are two points S and T that are symmetrical about the straight line M,

It is proved that when AC is not perpendicular to F 1F2, except for A and C, there are no two points S and T which are symmetrical about the straight line M. 。

Suppose there are two different points S(x3, y3), T(x4, y4),

Let the midpoint of ST be H(m, n), then kOH? kST=- 14，kOMkAC=- 14，

∴kOH=kOM=- 14k，

∴ The straight line m passes through the origin with a slope of-14k ≦-1k.

∴ If the hypothesis is not established,

When ∴AC is not perpendicular to F 1F2, there are no two points symmetrical about the straight line m except A and C. 。