It’s called a Diophantine Equation, and it’s sometimes known as the “summing of three cubes”.
A Diophantine equation is a polynomial equation, usually in two or more unknowns, such that only the integer solutions are sought or studied (an integer solution is such that all the unknowns take integer values).
It seems easy for $x^3+y^3+z^3=8$. $x=1$, $y=-1$ and $z=2.$
But what for higher values of $k$?
Using positive integers only for $x,y,z$ there are no solutions for $k<3$ but here are $3\le k \le 100$. The use of non-positive integers follows.
(x,y,z,k)
(1,1,1,3)
(1,1,2,10)
(1,2,2,17)
(2,2,2,24)
(1,1,3,29)
(1,2,3,36)
(2,2,3,43)
(1,3,3,55)
(2,3,3,62)
(1,1,4,66)
(1,2,4,73)
(2,2,4,80)
(3,3,3,81)
(1,3,4,92)
(2,3,4,99)
It appears that at least one of x,y,z must be positive if k is positive.
(x,y,z,k)
(-100,1,100,1)
(-99,1,99,1)
(-98,1,98,1)
(-97,1,97,1)
(-96,1,96,1)
(-95,1,95,1)
(-94,1,94,1)
(-93,1,93,1)
(-92,1,92,1)
(-91,1,91,1)
(-90,1,90,1)
(-89,1,89,1)
(-88,1,88,1)
(-87,1,87,1)
(-86,1,86,1)
(-85,1,85,1)
(-84,1,84,1)
(-83,1,83,1)
(-82,1,82,1)
(-81,1,81,1)
(-80,1,80,1)
(-79,1,79,1)
(-78,1,78,1)
(-77,1,77,1)
(-76,1,76,1)
(-75,1,75,1)
(-74,1,74,1)
(-73,1,73,1)
(-72,1,72,1)
(-71,1,71,1)
(-70,1,70,1)
(-69,1,69,1)
(-68,1,68,1)
(-67,1,67,1)
(-66,1,66,1)
(-65,1,65,1)
(-64,1,64,1)
(-63,1,63,1)
(-62,1,62,1)
(-61,1,61,1)
(-60,1,60,1)
(-59,1,59,1)
(-58,1,58,1)
(-57,1,57,1)
(-56,1,56,1)
(-55,1,55,1)
(-54,1,54,1)
(-53,1,53,1)
(-52,1,52,1)
(-51,1,51,1)
(-50,1,50,1)
(-49,1,49,1)
(-48,1,48,1)
(-47,1,47,1)
(-46,1,46,1)
(-45,1,45,1)
(-44,1,44,1)
(-43,1,43,1)
(-42,1,42,1)
(-41,1,41,1)
(-40,1,40,1)
(-39,1,39,1)
(-38,1,38,1)
(-37,1,37,1)
(-36,1,36,1)
(-35,1,35,1)
(-34,1,34,1)
(-33,1,33,1)
(-32,1,32,1)
(-31,1,31,1)
(-30,1,30,1)
(-29,1,29,1)
(-28,1,28,1)
(-27,1,27,1)
(-26,1,26,1)
(-25,1,25,1)
(-24,1,24,1)
(-23,1,23,1)
(-22,1,22,1)
(-21,1,21,1)
(-20,1,20,1)
(-19,1,19,1)
(-18,1,18,1)
(-17,1,17,1)
(-16,1,16,1)
(-15,1,15,1)
(-14,1,14,1)
(-13,1,13,1)
(-12,1,12,1)
(-12,9,10,1)
(-11,1,11,1)
(-10,1,10,1)
(-9,1,9,1)
(-8,-6,9,1)
(-8,1,8,1)
(-7,1,7,1)
(-6,1,6,1)
(-5,1,5,1)
(-4,1,4,1)
(-3,1,3,1)
(-2,1,2,1)
(-1,1,1,1)
(0,0,1,1)
(-47,-24,49,2)
(-6,-5,7,2)
(0,1,1,2)
(-5,4,4,3)
(1,1,1,3)
(-58,-43,65,6)
(-1,-1,2,6)
(-1,0,2,7)
(-100,2,100,8)
(-99,2,99,8)
(-98,2,98,8)
(-97,2,97,8)
(-96,2,96,8)
(-95,2,95,8)
(-94,2,94,8)
(-93,2,93,8)
(-92,2,92,8)
(-91,2,91,8)
(-90,2,90,8)
(-89,2,89,8)
(-89,41,86,8)
(-88,2,88,8)
(-87,2,87,8)
(-86,2,86,8)
(-85,2,85,8)
(-84,2,84,8)
(-83,2,83,8)
(-82,2,82,8)
(-81,2,81,8)
(-80,2,80,8)
(-79,2,79,8)
(-78,2,78,8)
(-77,2,77,8)
(-76,2,76,8)
(-75,2,75,8)
(-74,2,74,8)
(-73,2,73,8)
(-72,2,72,8)
(-71,2,71,8)
(-70,2,70,8)
(-69,2,69,8)
(-68,2,68,8)
(-67,2,67,8)
(-66,2,66,8)
(-65,2,65,8)
(-64,2,64,8)
(-63,2,63,8)
(-62,2,62,8)
(-61,2,61,8)
(-60,2,60,8)
(-59,2,59,8)
(-58,2,58,8)
(-57,2,57,8)
(-56,2,56,8)
(-55,2,55,8)
(-54,2,54,8)
(-53,2,53,8)
(-52,2,52,8)
(-51,2,51,8)
(-50,2,50,8)
(-49,2,49,8)
(-48,2,48,8)
(-47,2,47,8)
(-46,2,46,8)
(-45,2,45,8)
(-44,2,44,8)
(-43,2,43,8)
(-42,2,42,8)
(-41,2,41,8)
(-40,-17,41,8)
(-40,2,40,8)
(-39,2,39,8)
(-38,2,38,8)
(-37,2,37,8)
(-36,2,36,8)
(-35,2,35,8)
(-34,2,34,8)
(-34,15,33,8)
(-33,2,33,8)
(-32,2,32,8)
(-31,2,31,8)
(-30,2,30,8)
(-29,2,29,8)
(-28,2,28,8)
(-27,2,27,8)
(-26,2,26,8)
(-25,2,25,8)
(-24,2,24,8)
(-24,18,20,8)
(-23,2,23,8)
(-22,2,22,8)
(-21,2,21,8)
(-20,2,20,8)
(-19,2,19,8)
(-18,2,18,8)
(-17,2,17,8)
(-16,-12,18,8)
(-16,2,16,8)
(-16,9,15,8)
(-15,2,15,8)
(-14,2,14,8)
(-13,2,13,8)
(-12,2,12,8)
(-11,2,11,8)
(-10,2,10,8)
(-9,2,9,8)
(-8,2,8,8)
(-7,2,7,8)
(-6,2,6,8)
(-5,2,5,8)
(-4,2,4,8)
(-3,2,3,8)
(-2,2,2,8)
(-1,1,2,8)
(0,0,2,8)
(0,1,2,9)
(-3,-3,4,10)
(1,1,2,10)
(-2,-2,3,11)
(-11,7,10,12)
(-46,23,44,15)
(-1,2,2,15)
(-94,-48,98,16)
(-12,-10,14,16)
(0,2,2,16)
(-52,25,50,17)
(1,2,2,17)
(-2,-1,3,18)
(-95,47,91,19)
(-77,26,76,19)
(-16,-14,19,19)
(-2,0,3,19)
(-56,21,55,20)
(-2,1,3,20)
(-86,28,85,21)
(-14,-11,16,21)
(-10,8,8,24)
(2,2,2,24)
(-1,-1,3,25)
(-1,0,3,26)
(-100,3,100,27)
(-99,3,99,27)
(-98,3,98,27)
(-97,3,97,27)
(-96,3,96,27)
(-95,3,95,27)
(-94,3,94,27)
(-93,3,93,27)
(-92,3,92,27)
(-91,3,91,27)
(-90,3,90,27)
(-89,3,89,27)
(-88,3,88,27)
(-87,3,87,27)
(-86,3,86,27)
(-85,3,85,27)
(-84,3,84,27)
(-83,3,83,27)
(-82,3,82,27)
(-81,3,81,27)
(-80,3,80,27)
(-79,3,79,27)
(-78,3,78,27)
(-77,3,77,27)
(-76,3,76,27)
(-75,3,75,27)
(-74,3,74,27)
(-73,3,73,27)
(-72,3,72,27)
(-71,3,71,27)
(-70,3,70,27)
(-69,3,69,27)
(-68,3,68,27)
(-67,3,67,27)
(-66,3,66,27)
(-65,3,65,27)
(-64,3,64,27)
(-63,3,63,27)
(-62,3,62,27)
(-61,3,61,27)
(-60,3,60,27)
(-60,22,59,27)
(-59,3,59,27)
(-58,3,58,27)
(-57,3,57,27)
(-56,3,56,27)
(-55,3,55,27)
(-54,3,54,27)
(-53,3,53,27)
(-52,3,52,27)
(-51,3,51,27)
(-50,3,50,27)
(-49,3,49,27)
(-48,3,48,27)
(-47,3,47,27)
(-46,3,46,27)
(-45,3,45,27)
(-44,3,44,27)
(-43,3,43,27)
(-42,3,42,27)
(-41,3,41,27)
(-40,3,40,27)
(-39,3,39,27)
(-38,3,38,27)
(-37,-36,46,27)
(-37,3,37,27)
(-36,3,36,27)
(-36,27,30,27)
(-35,3,35,27)
(-34,3,34,27)
(-33,3,33,27)
(-32,3,32,27)
(-31,3,31,27)
(-30,3,30,27)
(-29,3,29,27)
(-28,3,28,27)
(-27,3,27,27)
(-26,3,26,27)
(-25,3,25,27)
(-24,-18,27,27)
(-24,3,24,27)
(-23,3,23,27)
(-22,3,22,27)
(-21,3,21,27)
(-20,3,20,27)
(-19,3,19,27)
(-18,-10,19,27)
(-18,3,18,27)
(-17,3,17,27)
(-16,3,16,27)
(-15,3,15,27)
(-14,3,14,27)
(-13,3,13,27)
(-12,3,12,27)
(-11,3,11,27)
(-10,3,10,27)
(-9,3,9,27)
(-8,3,8,27)
(-7,3,7,27)
(-6,3,6,27)
(-5,-4,6,27)
(-5,3,5,27)
(-4,3,4,27)
(-3,3,3,27)
(-2,2,3,27)
(-1,1,3,27)
(0,0,3,27)
(-59,31,56,28)
(-17,13,14,28)
(0,1,3,28)
(-20,13,18,29)
(-3,-2,4,29)
(1,1,3,29)
(-6,5,5,34)
(-4,-3,5,34)
(-1,2,3,34)
(-13,-8,14,35)
(0,2,3,35)
(-75,40,71,36)
(-3,-1,4,36)
(1,2,3,36)
(-56,37,50,37)
(-3,0,4,37)
(-27,16,25,38)
(-3,1,4,38)
(-52,20,51,43)
(-13,8,12,43)
(-7,-7,9,43)
(2,2,3,43)
(-7,-5,8,44)
(-3,2,4,45)
(-29,19,26,46)
(-2,3,3,46)
(-50,-50,63,47)
(-30,-14,31,47)
(-8,6,7,47)
(-26,-23,31,48)
(-2,-2,4,48)
(-4,-2,5,53)
(-1,3,3,53)
(-18,-15,21,54)
(-11,-7,12,54)
(0,3,3,54)
(-23,-23,29,55)
(-9,-6,10,55)
(-2,-1,4,55)
(1,3,3,55)
(-47,31,42,56)
(-21,-11,22,56)
(-2,0,4,56)
(-38,25,34,57)
(-2,1,4,57)
(-4,-1,5,60)
(-4,0,5,61)
(-43,22,41,62)
(-34,27,27,62)
(-4,1,5,62)
(-1,-1,4,62)
(2,3,3,62)
(-63,-37,67,63)
(-58,-38,63,63)
(-6,-4,7,63)
(-1,0,4,63)
(-100,4,100,64)
(-99,4,99,64)
(-98,4,98,64)
(-97,4,97,64)
(-96,4,96,64)
(-95,4,95,64)
(-94,4,94,64)
(-93,4,93,64)
(-92,4,92,64)
(-91,4,91,64)
(-90,4,90,64)
(-89,4,89,64)
(-88,4,88,64)
(-87,4,87,64)
(-86,4,86,64)
(-85,4,85,64)
(-84,4,84,64)
(-83,4,83,64)
(-82,4,82,64)
(-81,4,81,64)
(-80,-34,82,64)
(-80,4,80,64)
(-79,4,79,64)
(-78,4,78,64)
(-77,4,77,64)
(-76,4,76,64)
(-75,4,75,64)
(-74,4,74,64)
(-73,4,73,64)
(-72,4,72,64)
(-71,4,71,64)
(-70,4,70,64)
(-69,4,69,64)
(-68,4,68,64)
(-68,30,66,64)
(-67,4,67,64)
(-66,4,66,64)
(-65,4,65,64)
(-64,4,64,64)
(-63,4,63,64)
(-62,4,62,64)
(-61,4,61,64)
(-60,4,60,64)
(-59,4,59,64)
(-58,4,58,64)
(-57,4,57,64)
(-56,4,56,64)
(-55,4,55,64)
(-54,4,54,64)
(-53,4,53,64)
(-52,4,52,64)
(-51,4,51,64)
(-50,4,50,64)
(-49,4,49,64)
(-48,4,48,64)
(-48,36,40,64)
(-47,4,47,64)
(-46,4,46,64)
(-45,4,45,64)
(-44,4,44,64)
(-43,4,43,64)
(-42,4,42,64)
(-41,4,41,64)
(-40,4,40,64)
(-39,4,39,64)
(-38,4,38,64)
(-37,4,37,64)
(-36,4,36,64)
(-35,4,35,64)
(-34,4,34,64)
(-33,4,33,64)
(-32,-24,36,64)
(-32,4,32,64)
(-32,18,30,64)
(-31,4,31,64)
(-30,4,30,64)
(-29,4,29,64)
(-28,4,28,64)
(-27,4,27,64)
(-26,4,26,64)
(-25,4,25,64)
(-24,4,24,64)
(-23,4,23,64)
(-22,-17,25,64)
(-22,4,22,64)
(-21,4,21,64)
(-20,4,20,64)
(-19,4,19,64)
(-18,4,18,64)
(-17,4,17,64)
(-16,4,16,64)
(-15,4,15,64)
(-14,4,14,64)
(-13,4,13,64)
(-12,4,12,64)
(-11,4,11,64)
(-10,4,10,64)
(-9,4,9,64)
(-8,4,8,64)
(-7,4,7,64)
(-6,4,6,64)
(-5,-3,6,64)
(-5,4,5,64)
(-4,4,4,64)
(-3,3,4,64)
(-2,2,4,64)
(-1,1,4,64)
(0,0,4,64)
(0,1,4,65)
(1,1,4,66)
(-22,-19,26,69)
(-4,2,5,69)
(-64,23,63,70)
(-21,11,20,70)
(-33,-22,36,71)
(-24,12,23,71)
(-20,-16,23,71)
(-3,-3,5,71)
(-1,2,4,71)
(-27,-13,28,72)
(-10,7,9,72)
(0,2,4,72)
(-47,29,43,73)
(-24,-12,25,73)
(1,2,4,73)
(-55,26,53,78)
(-66,-49,74,79)
(-33,-19,35,79)
(-6,-6,8,80)
(2,2,4,80)
(-18,10,17,81)
(-15,12,12,81)
(3,3,3,81)
(-11,-11,14,82)
(-36,-32,43,83)
(-29,22,24,83)
(-23,-15,25,83)
(-5,-2,6,83)
(-2,3,4,83)
(-16,-9,17,88)
(-4,-4,6,88)
(-4,3,5,88)
(-7,6,6,89)
(-100,31,99,90)
(-73,-32,75,90)
(-51,-35,56,90)
(-48,19,47,90)
(-27,13,26,90)
(-9,-8,11,90)
(-5,-1,6,90)
(-3,-2,5,90)
(-1,3,4,90)
(-5,0,6,91)
(0,3,4,91)
(-8,-5,9,92)
(-5,1,6,92)
(1,3,4,92)
(-5,-5,7,93)
(-22,14,20,96)
(-22,17,18,97)
(-3,-1,5,97)
(-15,9,14,98)
(-3,0,5,98)
(-37,16,36,99)
(-5,2,6,99)
(-3,1,5,99)
(2,3,4,99)
(-6,-3,7,100)
This is the low-tech approach I took for this question. Note the optional output file in lines 140 and 220. The user may copy/paste or import the data into a spreadsheet for sorting as I did for the other answer.
100 print "Enter limit";
110 input l1
120 print time$
130 print
140 rem open "outfile.txt" for output as #1
150 for n1 = -l1 to l1
160 for n2 = n1 to l1
170 for n3 = n2 to l1
180 t0 = n1^3+n2^3+n3^3
190 for t9 = 1 to l1
200 if t0 = t9
210 print " " n1 n2 n3 t9 "."
220 rem print #1, n1 n2 n3 t9 "."
230 endif
240 next t9
250 next n3
260 next n2
270 rem The following IF/ELSE skips zero
280 rem for the next iteration of n1
290 if n1 = -1
300 n1 = 1
310 goto 160
320 endif
330 next n1
340 print time$