1.Introduction

About simultaneous equation {x^2+y^2+z^2=u^2, x^3+y^3+z^3=v^3}.
We show this equation has infinitely many integer solutions.

2.Theorem
     
A simultaneous equation {x^2+y^2+z^2=u^2, x^3+y^3+z^3=v^3} has infinitely many integer solutions.
 
Proof.

x^2+y^2+z^2=u^2..............................................................(1)
x^3+y^3+z^3=v^3..............................................................(2)

First, we obtain a parametric solution of equation (2) as below.

x = (-q^2-pq)a^2+(r^2-s^2)ba+(-pr+ps)b^2
y = (-p^2-pq)a^2+(-r^2+s^2)ba+(-qr+qs)b^2
z = (-pr-qr)a^2+(p^2-q^2)ba+(-s^2+rs)b^2
v = (-ps-qs)a^2+(p^2-q^2)ba+(r^2-rs)b^2

where p^3+q^3+r^3=s^3.
{a, b} is arbitrary.

Set (p,q,r,s)=(10,-24,27,19).
Substitute above parametric solution to equation (1) then (1) becomes to equation (3).

v^2 = 275380a^4-710192a^3b+719856a^2b^2-344896b^3a+66368b^4...................(3)

Set U = a/b, then V = v.
V^2 = 275380U^4-710192U^3+719856U^2-344896U+66368.............................(4)
    
This quartic equation (4) is birationally equivalent to an elliptic curve below.
Y^2 = X^3-X^2-218307X+ 18535352...............................................(5)

By using Mwrank in Sage, we obtain the rank=2 and generator is P1(1576 , 59104), P2(1853336/289, 2500563356/4913) on (5).

Hence, elliptic curve (5) has infinitely many rational solutions.

Transformation is given,

U = (11956X+3538688+150Y)/(375Y+13015X+5394872)
V = (-149631631296Y+6213326400X^2+2132435896152X-827478426801600+10125000X^3)/((375Y+13015X+5394872)^2)
X = (225V+51832-116588U+68770U^2)/(100U^2-80U+16)
Y = (35868V+9017632-32505768U+41667540U^2-39045VU-19127050U^3)/(500U^3-600U^2+240U-32).

Small solutions are shown below.
      [x , y , z]
P1    [46458, 32194, 6213]
P2    [-32194, -46458, -6213]
P1+P2 [70, 14, 23]

(x,y,z)=(70, 14, 23) is a smallest solution.

Q.E.D.