// The function NonADE takes a ring R and a squarefree homogeneous polynomial
// in three variables F of even degree d with coefficients in R,
// and it returns the non-ADE singular R-points of the projective
// surface S defined as the double cover of IP^2 ramified above the curve
// defined by F.

function NonADE(R,F)

// Check that the input is in the correct form
assert IsHomogeneous(F);
assert IsEven(Degree(F));
assert SquarefreePart(F) eq F;

// Define base ring.
A:=BaseRing(R);

// Get the (even) polynomial degree d of F and compute d/2
d:=Degree(F);
dd:=Numerator(d/2);

// Define ambient space.
PP<x,y,z,w>:=WeightedProjectiveSpace(A,[1,1,1,dd]);


// View double cover as hypersurface in weighted projective space.
FF:=w^2-Evaluate(F,[x,y,z]);
S:=Surface(PP,FF);

// Compute the singular locus of S.
SingS:=SingularSubscheme(S);

// Consistency check: Check that S has no non-isolated singularities.
// For surfaces, F being squarefree is equivalent to S:={w^2-F=0} having only isolated singularities.
assert Dimension(SingS) eq 0;

// Define list to store non-ADE singular points.
SP:=[];

// Consider all singular points of the affine patch x=1 of S, and store all
// non-ADE singularities in the list SP.
Sx:=AffinePatch(S,4);
SxSP:=SingularPoints(Sx);
for P in SxSP do
  if not IsSimpleSurfaceSingularity(P) then
    Append(~SP,S!P);
  end if;
end for;

// Consider all singular points of the affine patch y=1 of S, and store all
// non-ADE singularities in the list SP.
Sy:=AffinePatch(S,3);
SySP:=SingularPoints(Sy);
for P in SySP do
  if not IsSimpleSurfaceSingularity(P) then
    Append(~SP,S!P);
  end if;
end for;

// Consider all singular points of the affine patch z=1 of S, and store all
// non-ADE singularities in the list SP.
Sz:=AffinePatch(S,2);
SzSP:=SingularPoints(Sz);
for P in SzSP do
   if not IsSimpleSurfaceSingularity(P) then
    Append(~SP,S!P);
  end if;
end for;

// Remove duplicate entries from SP
SP:=[P : P in Set(SP)];

// Return all non-ADE singularities as a list of points or return.
// If S is smooth or has at most ADE singularities, return the empty list.
return SP;

end function;