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The following code finds the two solutions in the range of natural numbers $<11\cdot 31$ in a relatively quick manner.

def PossibilitiesMod( p, ordered=True ):
    """Compute all relevant (a,b,c,d) mod p
    """
    F = GF( p )
    powers = list( set( [ a^5 for a in F ] ) )

    possibilities = [ (a, b, c, d) 
                      for a in F for b in F for c in F for d in F
                      if a^5 + b^5 + c^5 + d^5 in powers ]
    if ordered:
        possibilities = [ (a, b, c, d) 
                          for (a, b, c, d) in possibilities
                          if a.lift() <= b.lift() <= c.lift() <= d.lift() ]
    possibilities.sort()
    return possibilities

p1 = 11      ; p2 = 31      ;
F1 = GF( p1 ); F2 = GF( p2 );

matchList1 = PossibilitiesMod( p1, ordered=False )
matchList2 = PossibilitiesMod( p2, ordered=True  )

lift_dic = {}
for x in [ 1 .. p1*p2 ]:
    lift_dic[ ( F1(x), F2(x) ) ] = x

fifthPowers = [ a^5 for a in range( 1, ( p1 * p2 *5^(1/5) ) .n() ) ]
solutions   = []

count = 0
for a1, b1, c1, d1 in matchList1:
    count += 1 
    if count % 1000 == 0:    print count
    for a2, b2, c2, d2 in matchList2:

        a = lift_dic[ ( a1, a2 ) ]
        b = lift_dic[ ( b1, b2 ) ]
        c = lift_dic[ ( c1, c2 ) ]
        d = lift_dic[ ( d1, d2 ) ]

        if a^5 + b^5 + c^5 + d^5 in fifthPowers:
            print "\n\n\n!!! SOLUTION !!! ( %s , %s , %s , %s )\n\n\n" % ( a, b, c, d )
            solutions.append( (a,b,c,d) )

Namely:

( 220 , 168 , 266 , 54 )
( 133 , 110 , 84 , 27 )

The last one is of course:

Lander and Parkin, 1966, "a direct search on the CDC 6600..."

(The first solution above is "twice the second one". Sorry.) To have a feasible search we have restricted first to relevant matches modulo some prime numbers, in our case $11$ and $31$. (Chosen so that the Euler indicator function on them delivers multiples of $5$, the power appearing in the equation.)

For the other related problems with higher powers this is of course too naive and slow...

https://en.wikipedia.org/wiki/Euler's_sum_of_powers_conjecture

http://www.cs.man.ac.uk/~rizos/EqualSums/equalsums.html

http://euler.free.fr/index.htm

(I'm afraid, this answers an other question, not explicitly the one posted above. The "no-if-restriction" is in m.h.o. a good joke, i've got the same feeling as facing "show this inequality without using algebra" some decades ago.)