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Let P(X,Y,Z) be a homogeneous polynomial in ℂ[X,Y,Z] whose locus M in ℂℙ2 is a nonsingular curve of genus ≥ 2.

Define M to be maximally symmetric if the following is not true:

There exists a continuous family { Pt   |  t ∊ [0,1] } of homogeneous polynomials in ℂ[X,Y,Z] such that 1), 2), and 3) hold:

1) P0 = P.

2) The locus Mt in ℂℙ2 of each Pt is nonsingular.

3) There is a group G such that the ambient isometry groups
Gt := IsomA(Mt) are all isomorphic to G for 0 <= ≤ t < 1, but G1 contains G as a proper subgroup.

Here the "ambient isometry group" IsomA(Mt) of a projective curve M in ℂℙ2 means the subgroup of Isom(ℂℙ2) = PSU(3) that carries M to itself.

Question: I'd like pointers to the literature regarding what may be known about a classification of such "maximally symmetric" projective curves up to ambient isometry, their defining polynomials, and especially their ambient isometry groups.
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Maximally symmetric smooth projective varieties in CP^2

Let P(X,Y,Z) be a homogeneous polynomial in ℂ[X,Y,Z] whose locus M in ℂℙ2 is a nonsingular curve of genus ≥ 2.

Define M to be maximally symmetric if the following is not true:

There exists a continuous family { Pt   |  t ∊ [0,1] } of homogeneous polynomials in ℂ[X,Y,Z] such that 1), 2), and 3) hold:

1) P0 = P.

2) The locus M in ℂℙ2 of each Pt is nonsingular.

3) There is a group G such that the ambient isometry groups
Gt := IsomA(Mt) are all isomorphic to G for 0 <= t < 1, but G1 contains G as a proper subgroup.

Here the "ambient isometry group" IsomA(Mt) of a projective curve M in ℂℙ2 means the subgroup of Isom(ℂℙ2) = PSU(3) that carries M to itself.

Question: I'd like pointers to the literature regarding what may be known about a classification of such "maximally symmetric" projective curves up to ambient isometry, their defining polynomials, and especially their ambient isometry groups.