In this work we analyze the eigendecomposition of the Hessian matrix of the Robin function R(x) for the spectral fractional Laplacian in orthogonally invariant domains. We prove that, if Ω a smooth bounded convex domain invariant under the action of an orthogonal transformation O then, for t∈{a∈Ω:O(a)=a}
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, the gradient vector
∇R(t)
is an eigenvector of the Jacobian
DO
associated to the eigenvalue
1
. Moreover, if
Ω
is a domain invariant under the reflection about a hyperplane
πv={x∈RN:x⋅v=0}
, there exists
η>0
such that
H(t)v=ηv
where
H
denotes the Hessian matrix of
R(x)
. Consequently, if
Ω
is invariant under the reflection about the hyperplanes
πvi
for a linearly independent set
{v1,…,vN}
, then the origin is a non degenerate critical point of
R(x)
. A short proof of the Brezis-Peletier-like formulas for
∇R
is also provided which allows us to prove the former results for any