Well, I think we have two totally different representations.
The contourplot asks for all momenta {x,y,z} with a given energie (1.6 here , Gianluca looked at energy = 1 )
ContourPlot3D[ 1.6 == 6 x^2 + 3 y^2 + z^2, {x, -1, 1}, {y, -1, 1}, {z, -1.3, 1.3}]
(or
RegionPlot3D[ 0 < 6 x^2 + 3 y^2 + z^2 < 1.6, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}]
)
But the momenta giving this energy don't have all the length 1. E.g. chose a momentum { .2, .3, z }
and ask, for which z the energy 1.6 is achieved . You will get a solution, but this has not the length 1.
Clear[x, y, z]
vec = {.2, .3, z} /. Solve[1.6 == 6 x^2 + 3 y^2 + z^2 /. x -> .2 /. y -> .3, z][[1, 1]]
vec.vec
If on the other hand you ask for the energies produced by a momentum of length 1 you have that these energies are angle-dependent.
Plot[6 x1[1, Pi/2, \[Phi]]^2 + 3 x2[1, Pi/2, \[Phi]]^2 + 1 x3[1, Pi/2, \[Phi]]^2, {\[Phi], 0, 2 \[Pi]}]
Plot[6 x1[1, th, 0]^2 + 3 x2[1, th, 0]^2 + 1 x3[1, th, 0]^2, {th, 0, Pi}]
This explains the first figure in blue: it is the energy given by momenta of length 1 pointing in different directions.
