(*Scaling sigma_x and sigma_z of the Pauli Spin matrices with n=1/2 give us the lemniscate. This image is used to produce
Lemniscate1.eps .  Also this verifies that the index is one
inside each lobe.  *)

n=1/2;
sigma1 = {{0,1}, {1,0}};
sigma2 = {{0,-I},{I,0}};
sigma3 = {{1,0},{0,-1}};
loclzr = KroneckerProduct[sigma1, n*sigma1 - x*IdentityMatrix[2]] + 
KroneckerProduct[sigma2, sigma2-y*IdentityMatrix[2]]+ 
KroneckerProduct[sigma3, n*sigma3-z*IdentityMatrix[2]] ;
MatrixForm[loclzr]
charpoly = FullSimplify[Det[loclzr]]
pl = ContourPlot3D[charpoly== 0, {x,-1,1}, {y,-1, 1}, {z, -1, 1},Contours->{{1,LightBlue}},PlotPoints->100,ViewPoint->{72,-21,9}]

Export["Lemniscate1.eps",pl,ImageSize->3.2*72]

A = ReplaceAll[loclzr,{x->0,y->1/2,z->0}]

MatrixForm[A]

Eigenvalues[A]

B = ReplaceAll[loclzr,{x->0,y->-1/2,z->0}]

MatrixForm[B]

Factor[CharacteristicPolynomial[A,x]]

Eigenvalues[B]

Factor[CharacteristicPolynomial[B,x]]