For what integer $n$ are there infinitely many $-a+nb+c = -d+ne+f$ where $a^6+b^6+c^6 = d^6+e^6+f^6$? (Much revised for clarity.) I was considering the system of equations,
$$-a+nb+c = -d+ne+f\tag1$$
$$a+b+c = d+e+f\tag2$$
$$a^2+b^2+c^2 = d^2+e^2+f^2\tag3$$
$$a^6+b^6+c^6 = d^6+e^6+f^6\tag4$$
Question 1. Is it true that, for a fixed integer $n$, if the system has an infinite number of co-prime integer solutions, then $n$ is a multiple of $3$?
Method: Eqs $(2)$ and $(3)$ can easily be given a complete solution. Incorporating $(1)$, I got,
$$(-2 p + \alpha q -\beta u)^k + (\beta  p - 2 u)^k+(\beta  q + \alpha  u)^k =\\ (-2 p + \alpha q +\beta u)^k + (\beta  p + 2 u)^k+(\beta  q -\alpha  u)^k\tag5$$
where $\alpha = n+1,\;\beta = n-1$. It is also true for $k=6$ if there is $p,q,n$ such that,
$$Poly_1:= (-3+n)(5-2n+n^2)p + 4n(1+n^2)q$$
$$Poly_2:= (-3+n)(5-2n+n^2)p^3 + 2(5+11n-5n^2+n^3)p^2q - (5+7n+15n^2-3n^3)pq^2 + 4n(1+n^2)q^3$$
and,
$$\color{red}{-}Poly_1 Poly_2 = \text{square}\tag6$$
A trivial solution is $q = \frac{(3-n)p}{2n}$ which yields,
$$\color{red}{-}Poly_1 Poly_2 = \frac{(-9+n^2)^2(-1+n)^4p^4}{4n^2}$$
Example: Let $n=12$, then,
$$((-2 p + 13 q - 11 u)^k+(11 p - 2 u)^k+(11 q + 13 u)^k =\\(-2 p + 13 q + 11 u)^k+( 11 p + 2 u)^k+( 11 q - 13 u)^k\tag7$$
which is already true for $k=1,2$. But it is also for $k=6$ if,
$$225 p^3 + 458 p^2 q + 587 p q^2 + 1392 q^3 = -3 (75 p + 464 q) u^2\tag8$$ 
An initial point is $p,q = -104,17.$ Hence $(8)$ can be easily turned into an elliptic curve, so there is an infinite number of integer solutions to $(7)$. 
Question 2. For what other positive integer $n$ below a bound can we find solutions to non-zero $(6)$ or with $(3-n)p-2nq \neq 0$? (The constraint is to prevent trivial solutions. I have found $n=12, 15, 21, 30, 33, 135$ but I am not sure if this is exhaustive for $n<150$.) 

As asked by OP here are values of $n$ where the BSD Conjecture predicts a strictly positive rank for the elliptic curve
12,
13,
15,
16,
17,
18,
21,
23,
24,
25,
26,
27,
30,
32,
33,
35,
36,
38,
40,
41,
43,
44,
47,
49,
52,
53,
54,
56,
57,
64,
65,
66,
67,
69,
73,
75,
76,
78,
79,
80,
81,
85,
86,
88,
90,
91,
93,
94,
95,
97.
