How to find new parametric families for a diophantine equation $$A^4+28A^3B+70A^2B^2+28AB^3+B^4=C^4+28C^3D+70C^2D^2+28CD^3+D^4$$ Example smallest family:

$$A=n^3-2n^2+1$$ $$B=n^3+2n^2-1$$ $$C=n^3-n-1$$ $$D=n^3-n+1$$

it was found that there is a polynomial defining an even larger part of the paired solutions $A=10n^3+15n^2+7n+1$ However, it is not clear how its other parameters are related to it. Found 45 canonical nontrivial positive solutions in ranges [0;8000]

1 15 5 7   (family n=2)
5 4389 853 1877
15 671 167 295
23 459 51 367
23 5302 404 3651
33 95 59 61   (family n=4)
33 217 97 113 [Unknown family-?1,2]
33 437 135 227 [Unknown family-?1,2]
55 3401 797 1483
95 2567 719 1151
105 691 125 643
145 287 209 211   (family n=6)
145 1313 521 649
155 531 307 323 [Unknown family-?1,2]
155 869 413 467 [Unknown family-?1,2]
217 7975 2071 3529
287 6463 1903 2927
385 639 503 505   (family n=8)
385 4009 1513 1945
413 1045 685 701
413 3043 1303 1553
427 1573 883 937 [Unknown family-?1,2]
427 2235 1091 1219 [Unknown family-?1,2]
433 3537 1065 2273
801 1199 989 991   (family n=10)
855 1807 1279 1295
855 7337 2969 3655
909 3485 1925 2053 [Unknown family-?1,2]
909 4579 2269 2519 [Unknown family-?1,2]
1441 2015 1715 1717   (family n=12)
1479 3823 2007 3079
1529 2865 2137 2153
1595 3893 2591 2645
1639 5111 3071 3199
1661 6531 3571 3821 [Unknown family-?1,2]
1661 8165 4085 4517 [Unknown family-?1,2]
2353 3135 2729 2731   (family n=14)
2483 4267 3307 3323
2587 5605 3925 3979
2665 7161 4577 4705
2717 8947 5257 5507
2743 ????? ???? ????? [Theoretical solution-?1,2]
3585 4607 4079 4081   (family n=16)
3765 6061 4837 4853
5185 6479 5813 5815   (family n=18)
5423 8295 6775 6791

The purpose of the question is to obtain new parametric families for a given diophantine equation.

I don't speak English, please correct any mistakes if there are any.

How to find new parametric forms for the OP diophantine equation $$A^4+28A^3B+70A^2B^2+28AB^3+B^4=C^4+28C^3D+70C^2D^2+28CD^3+D^4$$ The smallest parametric solution can be found in this collection by Tito Piezas III.

$$A=n^3-2n^2+1$$ $$B=n^3+2n^2-1$$ $$C=n^3-n-1$$ $$D=n^3-n+1$$

By searching, I discovered a polynomial of the third degree defining at least two more parametric families. $A=10n^3+15n^2+7n+1$ I would like to mention Tomita, who found an interesting solution to the OP equation for the 2nd degree $$A=-3n^2-96n+576$$ $$B=-7n^2+224n-1216$$ $$C=-19n^2+224n-448$$ $$D=9n^2-96n-192$$

The purpose of the question is to obtain new parametric families for a given diophantine equation.

I don't speak English, please correct any mistakes if there are any.