A solution is two lattice points on one curve; the curves worth having live on a K3 surface, and they come in an elliptic pencil. Four pictures, following Brudno, Brudno–Kaplansky and Bremner.
The surface V
Write Q(s,t) = s² + st − t². Bremner's surface V ⊂ P⁵, with coordinates (x, y, z, u, v, w), is cut out by the three quadrics
Q(x,u) = Q(w,z), Q(y,v) = Q(u,x), Q(z,w) = Q(v,y).
It is a K3 surface: a smooth complete intersection of three quadrics in P⁵.
Why its points solve the equation
The cubing identity
Q(s,t)³ − Q(t,s)³ = 2 (s⁶ − t⁶)
turns the three quadric equations, after cubing, into x⁶ − u⁶ + y⁶ − v⁶ + z⁶ − w⁶ = 0; adding them as they stand gives x² − u² + y² − v² + z² − w² = 0 (the st terms cancel in pairs). So every rational point of V is a point of M₆ ∩ M₂: an equal sum of sixth powers that is also an equal sum of squares. Panel 1 shows this literally: the two triples are two points of the curve where the superellipsoid meets the sphere.
The Brudno pencil
Following Brudno–Kaplansky's substitution, put
y = a + (λ−1)b, z = (λ−1)a − b, v = −(λ−1)a − λb, w = λa − (λ−1)b.
For each λ this is a P³ with coordinates (x : u : a : b), and V ∩ P³ is the curve
Eλ: x² − u² = (λ²−1)(a² + b²), xu = λa² − (3λ²−5λ+3)ab − λb²,
an intersection of two quadrics in P³, i.e. an elliptic curve. The fibres Eλ sweep out V: this is an elliptic fibration π : V → P¹. Bremner's Weierstrass model is
y′² = x′³ − 2(5PQ − 2B²) x′² + 25P²Q² x′, B = λ²−1, P = 2λ²−2λ+1, Q = λ²−2λ+2,
with torsion point T = (5PQ, 10BPQ) of order 4 (2T = (0,0)) and Mordell–Weil generators
g₁ = (5P², 15BP²), g₂ = (5Q(2λ−1)², 10Q(λ−1)(2λ−1)(3λ²−λ+1)), g₃ = (5P(λ−2)², 10P(λ−1)(λ−2)(λ²−λ+3)).
The singular fibres are I₂ at λ = ±1, I₄ at the complex roots of P and of Q, and I₁ at the four complex roots of 9λ⁴ − 30λ³ + 47λ² − 30λ + 9. Over the reals every fibre is connected, which is why Panel 2 always shows a single closed loop. In the picture x′ is rescaled by 5PQ and then sent to x′/(1+x′), so that the whole real locus, including the point at infinity on the right edge, fits in a fixed frame.
Sections, and Brudno's quartic
A section is a rational curve S ⊂ V meeting every fibre once; it is the same thing as a point of E(Q(λ)) = ⟨g₁, g₂, g₃⟩ ⊕ ⟨T⟩, written
S = n₁g₁ + n₂g₂ + n₃g₃ + kT, ni ∈ Z, k ∈ Z/4.
Plugging λ = m/n into the coordinates of S and clearing denominators gives six binary forms in (m, n): a polynomial identity x⁶+y⁶+z⁶ = u⁶+v⁶+w⁶. Brudno's 1976 quartic is exactly the section (−1, 0, 1, 0) (up to the symmetries of V); Bremner's curves (ii), (iii), (iv), Choudhry's octic and Delorme's degree-9 curve are sections too, and the four remaining orbits of degree ≤ 12 (degrees 7, 10, 11, 11) are new. The eight lines Γi and eight of the conics are the sections of degree 1 and 2.
The degree formula
The degree of the section curve (as a curve in P⁵, the degree of the six forms) depends only on the lattice part:
deg S = 2n₁² + 4n₂² + 4n₃² − 2n₁n₂ − 2n₁n₃ + 2n₂n₃ + 2n₁ − 3n₂ − 3n₃ + 1 = 2⟨S − γ/4, S − γ/4⟩, γ = −g₁ + g₂ + g₃,
where ⟨ , ⟩ is the height pairing with Gram matrix [[1, −½, −½], [−½, 2, ½], [−½, ½, 2]] on (g₁, g₂, g₃). There is no torsion correction: S, S+T, S+2T, S+3T are symmetric images of one another. Since the quadratic form is positive definite there are finitely many sections of each degree; those of degree ≤ 12 are the 144 drawn in Panel 2 (36 lattice classes × 4 torsion translates, 12 orbits under the symmetry group), and there is none of degree 3 or 12.
Brudno–Kaplansky
Brudno's 1970 ansatz v = y − z, w = y + z is the case (p, q) = (0, 1) of the substitution above, i.e. the fibre λ = 0 (equivalently λ = ∞, 2/3, 3/2 under the Klein four-group of symmetries that act on λ). On it the generators specialise with a rank-2 kernel, and the surviving generator g₁ is the Brudno–Kaplansky point P of E₁: W² = (2T+1)(10T²+2T+1), which is the curve 480f4. The quartic z² = x⁴ + 38x²y² + y⁴ of 1970 is a different model, 2-isogenous (its Jacobian is 480f1), and its rational points correspond to the even multiples of P.
Data: the 74 primitive solutions with all terms ≤ 1343 (complete to that bound; exhaustive enumeration), the 144 sections of degree ≤ 12 computed exactly in Sage from Bremner's generators, and Bremner's map from Eλ to the Weierstrass model. The applet itself is one HTML file with no dependencies apart from KaTeX for typesetting; the arithmetic shown is exact (BigInt).