Near-misses to the Fermat quintic threefold $x_1^5+x_2^5+x_3^5+x_4^5+x_5^5=0$

The Fermat quintic threefold is given by the equation,

$$x_1^5+x_2^5+x_3^5+x_4^5+x_5^5=0\qquad\qquad$$

$\hskip1.5in$

(Incidentally, this threefold is a Calabi-Yau manifold, a type of manifold important to string theory.) In the integers, there are only four primitive solutions known, two which have the form,

$$27^5+ 84^5+ 110^5+ 133^5 - 144^5=0$$ $$55^5+ 3183^5+ 28969^5+ 85282^5 -85359^5=0$$

We ask for integer "near-misses" where the sum, instead of vanishing, is some small integer $N$. For this post, the special case $\color{blue}{N = 1}$. For example, Simon Goater found the near-miss,

$$645^5 + 1523^5 + 1722^5 + 2506^5- 2615^5 = 1$$

Since we allow negative terms but avoid $x_1^5+(-x_1)^5+\dots=1$, then there are $\color{blue}{8+10+1=19}$ "non-trivial" solutions known so far.

I. First kind

1 + 89^5 + 118^5 = 38^5 + 47^5 + 123^5 
1 + 127^5 + 430^5 = 16^5 + 310^5 + 412^5 
1 + 328^5 + 709^5 = 5^5 + 388^5 + 705^5 
1 + 588^5 + 772^5 = 59^5 + 511^5 + 791^5 
1 + 561^5 + 1151^5 = 401^5 + 616^5 + 1146^5 
1 + 1073^5 + 2297^5 = 379^5 + 686^5 + 2306^5 
1 + 4167^5 + 4283^5 = 1039^5 + 2601 + 4811^5 
1 + 4823^5 + 6377^5 = 1089^5 + 3501 + 6611^5 

These 8 are from Duncan Moore's $(5,3,3)$ database and there are apparently no more solutions with all positive terms $x_i< 50000$ (which seems strange).

II. Second kind

-1 + 541^5 + 551^5 = 151^5 + 317^5 + 623^5 
-1 + 571^5 + 675^5 = 63^5 + 475^5 + 707^5
-1 + 1654^5 + 1783^5 = 699^5 + 763^5 + 1974^5
-1 + 611^5 + 2651^5 = 781^5 + 882^5 + 2648^5
-1 + 2316^5 + 3249^5 = 1286^5 + 2029^5 + 3299^5 
-1 + 355^5 + 4573^5 = 1660^5 + 3903^5 + 4044^5  
-1 + 9317^5 + 10609^5 = 2937^5 + 5509^5 + 11479^5
-1 + 12776^5 + 15112^5 = 293^5 + 12237^5 + 15357^5 
-1 + 8183^5 + 17477^5 = 3208^5 + 5659^5 + 17542^5
-1 + 8451^5 + 24899^5 = 1333^5 + 7104^5 + 24912^5

The first 2 are from user wxffles while the remaining 8 with terms $x>1000$ are from Simon Goater. Except for the term $x_1 = -1$, I don't know if these 10 is the complete list with positive $x_i < 25000$.

III. Third kind

We define it as the form $x_1^5+x_2^5+x_3^5+x_4^5 = x_5^5 \pm 1$ and only one solution is known so far

645^5 + 1523^5 + 1722^5 + 2506^5 = 2615^5 + 1

also by Simon Goater. (Note: This distinction into three kinds is just partly aesthetic and not absolute.)

IV. Question

Like $x^3+y^3+z^3 = 1$, are there infinitely many integer solutions to $x_1^5+x_2^5+x_3^5+x_4^5+x_5^5 = 1$, perhaps using a polynomial parameterization? Or, if that is too difficult, what is the complete list of solutions with terms $|x_i|< 50000$ (or higher, if possible)?

P.S. This 2023 MSE post asks if we can solve the equation for any $N^5$ (true for $N = 1 \to 1000$), this 2026 MSE post asks for any $N$, this post by DanielV asks for the smallest unknown $N=7$, and this MO post asks for efficient methods to find these solutions.