Bug (NaNs) when differentiating eigenvectors of Symmetric matrices

Hello,
I noticed at a certain point my code started to crash because of some NaNs introduced by Forward diff when performing derivatives of the eigenvectors of matrix.

Here was my previous code (it was working at a certain point):

 function buggy(x)
       mat = zeros(eltype(x),(2, 2))
       mat[1,1] = x[1]
       mat[1,2] = x[2]
       mat[2,1] = x[2]
       mat[2,2] = x[3]
       (ω, pols) = eigen(Symmetric(mat))
       return pols
end

When differentiating with:

julia> vv = [82.62687031649999 0.0 82.62687031649999]
julia> ForwardDiff.jacobian(buggy, vv)
4×3 Matrix{Float64}:
 NaN  NaN   NaN
 NaN  -Inf  NaN
 NaN   Inf  NaN
 NaN  NaN   NaN

However, if I remove the specification that the matrix must be Symmetric, then it works.

function good(x)
       mat = zeros(eltype(x),(2, 2))
       mat[1,1] = x[1]
       mat[1,2] = x[2]
       mat[2,1] = x[2]
       mat[2,2] = x[3]
       (ω, pols) = eigen(mat)
       return pols
end
julia> ForwardDiff.jacobian(good, vv)
4×3 Matrix{Float64}:
 0.0  0.0  0.0
 0.0  0.0  0.0
 0.0  0.0  0.0
 0.0  0.0  0.0

I remember I had to specify Symmetric because, at a certain point, ForwardDiff was working only with that (I do not remember exactly).

Reproduce the report with ForwardDiff.jacobian on the provided buggy function and input, comparing eigen(Symmetric(mat)) with eigen(mat). Read the eigenvector differentiation path involved by the Symmetric wrapper and determine what behavior should avoid the reported NaN and Inf values; done means the reproduction no longer produces those values.