I have encountered a case of a 3x3 symmetric matrix where 2 of the eigenvectors corresponding to 2 different eigenvalues are equal.
In order to find an orthogonal/orthonormal basis for the eigenspace all the vectors in this basis must be mutually orthogonal. However running Gram Schmidt on 2 equal vectors returns zero, which makes perfect sense because the vectors are linearly dependent of each other, well they are identical, in other words there doesn't exist an orthogonal component of a vector to itself.
We also know the basis must consist of 3 vectors, if I understand this correctly? So the question is how can one find a vector that is mutually orthogonal to the 2 others, but is still a "valid" eigenvector, would we pick any vector that is mutually orthogonal to the other 2 vectors? If yes, how can one do that easily?
Conclusion/correction: eigenvectors corresponding to different eigenvalues must be different (assuming we are not dealing with 0).
Different scenario where $x_1=x_2$ and $λ_1=λ_2$:
However let's assume , the basis for the eigenspace must still consist of 3 mutually orthogonal vectors right? So in that case how would one procede to find the 2nd basis-element for the eigenspace? If they are equal they can't be mutually orthogonal, but if they are mutually orthogonal in this case wouldn't they correspond to different eigenvalues? If I remember correctly we would get 2 free variables, thus 2 eigenvectors for the repeated eigenvalue (twice) as a result from the same RREF. In case these are not orthogonal we would use GS on one of them.