We searched the rational solutions of {x+y+z=n, x^3+y^3+z^3=n}.
Moreover, we show a table of smallest solutions for n<100.
Smallest solutions for n=75,89,and 93 are very large!

Method 

x+y+z=n,..................................................................(1)

x^3+y^3+z^3=n.............................................................(2)
                    
Substitute z=n-x-y to equation (2), then equation (2) becomes to equation (3).

(-3y+3n)x^2+(-3y^2-3n^2+6ny)x+n^3+3ny^2-3n^2y-n=0.........................(3)

To obtain the rational solution x of equation (3), the discriminant must be square as follows.

v^2 = 9y^4-18n^2y^2+(12n^3-12n)y-3(n^3-4n)n...............................(4)

This quartic equation (4) is birationally equivalent to an elliptic curve below.

Y^2 = X^3+36n^2X^2+(432n^4-432n^2)X+1296n^6-2592n^4+1296n^2...............(5)

We used Stoll's ratpoints and Pari-gp to find the rational points and calculate the rank.


Further, there are infinitely many parametric solutions of {x+y+z=3, x^3+y^3+z^3=3}.
We show two parametric solutions below.
It may be difficult to find the parametric solution for integral n except 3.

x = (2a^2+3ab-7a+3b^2-9b+8)/((b-1)(a-2+b))

y = (2b^2+3ab-7b-9a+3a^2+8)/((a-1)(a-2+b))

z = -(2a^2+ab-5a+2b^2-5b+5)/((b-1)(a-1))


x = -(2a^4-10a^3+2a^3b+23a^2+5a^2b^2-16a^2b-30a-28b^2a+6ab^3+44ab+18+41b^2-18b^3+3b^4-42b)/((b-1)(a-b)(a-2+b)(2a-3+b))

y = (3a^4+6a^3b-18a^3+5a^2b^2-28a^2b+41a^2-16b^2a+2ab^3-42a+44ab+2b^4+23b^2-30b-10b^3+18)/((a-1)(a-2+b)(a-b)(a-3+2b))

z = (2a^4+6a^3b-14a^3+11a^2b^2-40a^2b+41a^2+6ab^3-40b^2a-54a+80ab-14b^3+41b^2+2b^4-54b+27)/((b-1)(a-1)(2a-3+b)(a-3+2b))

a, b: arbitrary


Table 

   Search range: n<100
   [0,0,0]: Rank=0
   Table of smallest solutions  of {x+y+z=n, x^3+y^3+z^3=n}.
   
   n                               x                               y                               z
[  1,                              1,                              1,                             -1]
[  2,                              0,                              0,                              0]
[  3,                              1,                              1,                              1]
[  4,                              0,                              0,                              0]
[  5,                              0,                              0,                              0]
[  6,                              0,                              0,                              0]
[  7,                              0,                              0,                              0]
[  8,                            -16,                             15,                              9]
[  9,                              0,                              0,                              0]
[ 10,                           41/3,                            -17,                           40/3]
[ 11,                              0,                              0,                              0]
[ 12,                              0,                              0,                              0]
[ 13,                              0,                              0,                              0]
[ 14,                              0,                              0,                              0]
[ 15,                              0,                              0,                              0]
[ 16,                              0,                              0,                              0]
[ 17,                          153/5,                            -33,                           97/5]
[ 18,                          141/5,                            -32,                          109/5]
[ 19,                           95/3,                            -35,                           67/3]
[ 20,                            -56,                             55,                             21]
[ 21,                          118/3,                            -42,                           71/3]
[ 22,                          112/3,                            -41,                           77/3]
[ 23,                  -814325/16362,                  1679437/66458,                  1265791/26649]
[ 24,                              0,                              0,                              0]
[ 25,                        2025/17,                         -239/2,                         863/34]
[ 26,                              0,                              0,                              0]
[ 27,                        1025/11,                            -94,                         306/11]
[ 28,                            -59,                             56,                             31]
[ 29,                          261/2,                           -131,                           59/2]
[ 30,                              0,                              0,                              0]
[ 31,                              0,                              0,                              0]
[ 32,                   705862/15045,                   -329512/5959,                  1039982/25755]
[ 33,                              0,                              0,                              0]
[ 34,                           -120,                             35,                            119]
[ 35,                          105/2,                            -61,                           87/2]
[ 36,                            -75,                             71,                             40]
[ 37,                 -1068153/15652,                   677767/10972,                  2760915/63511]
[ 38,                          113/2,                            -66,                           95/2]
[ 39,                     -25013/333,                        3124/45,                       8268/185]
[ 40,                              0,                              0,                              0]
[ 41,                              0,                              0,                              0]
[ 42,                       -1133/15,                          151/3,                          336/5]
[ 43,                     -17849/150,                      12751/282,                    137213/1175]
[ 44,                              0,                              0,                              0]
[ 45,                              0,                              0,                              0]
[ 46,                              0,                              0,                              0]
[ 47,                           -141,                            139,                             49]
[ 48,                              0,                              0,                              0]
[ 49,                              0,                              0,                              0]
[ 50,                         -275/3,                             59,                          248/3]
[ 51,                         -263/3,                          391/6,                          147/2]
[ 52,                         -481/4,                            116,                          225/4]
[ 53,                        1317/17,                         -183/2,                        2279/34]
[ 54,                              0,                              0,                              0]
[ 55,                           -110,                            103,                             62]
[ 56,                          189/2,                           -104,                          131/2]
[ 57,                              0,                              0,                              0]
[ 58,                              0,                              0,                              0]
[ 59,                              0,                              0,                              0]
[ 60,                         3959/6,                           -660,                          361/6]
[ 61,                        2135/17,                         -395/3,                        3421/51]
[ 62,                              0,                              0,                              0]
[ 63,                              0,                              0,                              0]
[ 64,                         1102/3,                           -368,                          194/3]
[ 65,                           -111,                             91,                             85]
[ 66,                         1131/8,                         -442/3,                        1727/24]
[ 67,                              0,                              0,                              0]
[ 68,                              0,                              0,                              0]
[ 69,                      27589/201,                       -3040/21,                      35880/469]
[ 70,                         -358/3,                          581/6,                          185/2]
[ 71,              440814221/4301490,             -571125263/4677447,              158940077/1753870]
[ 72,                         -368/3,                             99,                          287/3]
[ 73,                         -373/3,                            100,                          292/3]
[ 74,                    -134347/705,                      66032/355,                   786439/10011]
[ 75,     -5825523201274/31516848333,    68175623690938/379451906721,    54530863942575/680192947477]
[ 76,                      -21109/18,                      17809/234,                       15244/13]
[ 77,                        5137/45,                         -668/5,                          868/9]
[ 78,                              0,                              0,                              0]
[ 79,                      -19829/37,                         1606/3,                       8834/111]
[ 80,                              0,                              0,                              0]
[ 81,                      63101/531,                       -8821/63,                      42201/413]
[ 82,                              0,                              0,                              0]
[ 83,                         -747/5,                            133,                          497/5]
[ 84,                              0,                              0,                              0]
[ 85,               -22702787/127281,              282219380/2994761,                30120148/178089]
[ 86,                         1763/6,                         -593/2,                          266/3]
[ 87,                              0,                              0,                              0]
[ 88,                              0,                              0,                              0]
[ 89,  -119240565206609/683118665882, 1128754699484631/6950084184559,  596611119161079/5898605038262]
[ 90,                              0,                              0,                              0]
[ 91,                              0,                              0,                              0]
[ 92,                   1101332/8695,                     -34801/222,                    172159/1410]
[ 93,       1886165593135/9336130662,   -89963670031473/428113062203,      2244480453379/22198103826]
[ 94,                          517/3,                           -185,                          320/3]
[ 95,                       -6245/29,                      44897/435,                        3107/15]
[ 96,                              0,                              0,                              0]
[ 97,                           -295,                            291,                            101]
[ 98,                              0,                              0,                              0]
[ 99,                           -176,                            155,                            120]