New multi-grade solutions $x_1^k+x_2^k+x_3^k=y_1^k+y_2^k+y_3^k=z_1^k+z_2^k+z_3^k, k<5$

For the next multigrade diophantine chain $$x_1^k+x_2^k+x_3^k=y_1^k+y_2^k+y_3^k=z_1^k+z_2^k+z_3^k=N, k<5$$ This question is partly inspired by this similar question, but here we will look at degrees that are less than 5, which makes the discovery of new multi-grade solutions more likely. $$\boxed{\begin{array} {|c|c|c|c|} \hline & {\color{blue}{k=4}} & {\color{orange}{k=3,4}} & {\color{blue}{k=3}} \\ \hline {\color{blue}{k=2}} & {\color{blue}{k=2,4}} & {\color{Goldenrod}{k=2,3,4}} & {\color{blue}{k=2,3}} \\ \hline {\color{blue}{k=1,2}} & {\color{blue}{k=1,2,4}} & {\color{red}{k=1,2,3,4}} & {\color{red}{k=1,2,3}} \\ \hline {\color{blue}{k=1}} & {\color{blue}{k=1,4}} & {\color{orange}{k=1,3,4}} & {\color{blue}{k=1,3}} \\ \hline \end{array}}$$

• The orange marks are existing solutions that are not triplets.
• The blue marks are the existing multi-grade chains
• The red marks indicate a complete lack of solutions.

Both examples were found thanks to Ajai Choudhry in 1991 and 2001 respectively. $$\small (-3254)^k+5583^k+5658^k=(-1329)^k+2578^k+6738^k, k=1,3,4$$ $$\small (-815)^k+358^k+1224^k=(-776)^k+(-410)^k+1233^k, k=2,3,4$$

• For the case $k=1,3,4$, we can find a parametric family to confirm an infinite set of solutions. But the question remains about the existence of multi-grade chains.
• There is also a second family of polynomials of degree 34, I will not be able to mark them in the post. $$ \scriptsize \begin{align*} x_1 &= p^{10} + 5 p^{9} q + 9 p^{8} q^{2} + 6 p^{7} q^{3} - 3 p^{6} q^{4} - 15 p^{5} q^{5} - 18 p^{4} q^{6} - 6 p^{3} q^{7} + 2 p^{2} q^{8} + p q^{9} \\ x_2 &= p^{9} q + 8 p^{8} q^{2} + 18 p^{7} q^{3} + 12 p^{6} q^{4} - 3 p^{5} q^{5} - 3 p^{4} q^{6} + 6 p^{3} q^{7} + 9 p^{2} q^{8} + 5 p q^{9} + q^{10} \\ x_3 &= - p^{9} q - 7 p^{8} q^{2} - 14 p^{7} q^{3} - 4 p^{6} q^{4} + 19 p^{5} q^{5} + 26 p^{4} q^{6} + 10 p^{3} q^{7} - p^{2} q^{8} - p q^{9} \\ y_1 &= p^{10} + 5 p^{9} q + 9 p^{8} q^{2} + 6 p^{7} q^{3} - 3 p^{6} q^{4} - 3 p^{5} q^{5} + 12 p^{4} q^{6} + 18 p^{3} q^{7} + 8 p^{2} q^{8} + p q^{9} \\ y_2 &= p^{9} q + 2 p^{8} q^{2} - 6 p^{7} q^{3} - 18 p^{6} q^{4} - 15 p^{5} q^{5} - 3 p^{4} q^{6} + 6 p^{3} q^{7} + 9 p^{2} q^{8} + 5 p q^{9} + q^{10} \\ y_3 &= - p^{9} q - p^{8} q^{2} + 10 p^{7} q^{3} + 26 p^{6} q^{4} + 19 p^{5} q^{5} - 4 p^{4} q^{6} - 14 p^{3} q^{7} - 7 p^{2} q^{8} - p q^{9} \\ \end{align*} $$

For the case $k=2,3$, Ajai Choundhry found a parametric representation in multi-grade form. Also in the article there was a mention about the parametric solution $k=2,3,4$, which I could not find, but it probably exists. I would be grateful for a link to it in the comments to help future researchers.

$$ \small \begin{align*} x_1 &= 3p^6 + 28p^5q + 78p^4q^2 + 48p^3q^3 + 468p^2q^4 + 1008pq^5 + 648q^6, \\ x_2 &= -3p^6 + 28p^5q - 78p^4q^2 + 48p^3q^3 - 468p^2q^4 + 1008pq^5 - 648q^6, \\ x_3 &= 36pq(p^2 + 2q^2)(p^2 + 18q^2), \\[1ex] y_1 &= -3p^6 + 28p^5q - 138p^4q^2 + 48p^3q^3 - 1476p^2q^4 + 1008pq^5 + 648q^6, \\ y_2 &= 3p^6 + 28p^5q + 138p^4q^2 + 48p^3q^3 + 1476p^2q^4 + 1008pq^5 - 648q^6, \\ y_3 &= -24pq(p^2 - 6q^2)(p^2 + 18q^2), \\[1ex] z_1 &= -3p^6 + 28p^5q + 246p^4q^2 + 48p^3q^3 + 828p^2q^4 + 1008pq^5 + 648q^6, \\ z_2 &= 3p^6 + 28p^5q - 246p^4q^2 + 48p^3q^3 - 828p^2q^4 + 1008pq^5 - 648q^6, \\ z_3 &= 72pq(p^2 - 6q^2)(p^2 + 2q^2) \end{align*} $$ $$\scriptsize 2281^k+(-113)^k+2052^k=115^k+2053^k+2280^k=2803^k+(-635)^k+(-1080)^k, k=2,3 $$

Well, as you can see, we still have 3 cases that stand out noticeably and illogically in the table, having a relationship with degrees $k =3,4$ which are represented in two ways, but nothing is known about the three ways. Negative integers are allowed. Non-triviality is 3 different ways to represent the number N for each degree. More information about the solutions can be found here

Question. Are there nontrivial chains for these, $k=3,4$ or maybe $k=2,3,4$ or $k=1,3,4?$ English is not my native language any corrections are welcome