The Diophantine equation $x_1^6+x_2^6+x_3^6=z^2$ where exactly one $(x_i)\equiv 0{\pmod 7}$.

Above equation shown below:

$x^6+y^6+z^6=w^2$

There are two more numerical solutions for $(x,y,z)< 5000$ $(x,y,z,w)=(2043,2184,2518,20883327517)$

$(x,y,z,w)=(3087,3404,4482,102604114673)$

Above solutions including numerical solution shown by "OP" $(x,y,z,w)=(140,213,390,60163597)$

have been arrived at through elliptical curve method by Seiji Tomita on his web site shown below:

  http://www.maroon.dti.ne.jp/fermat/eindex.html

Click on the above link & select 'Computional number theory" &

check out his articles # 166 and #167

Equation: $x^6+y^6+z^6=w^2$ ----(1)

Regarding the subsequent request by @OldPeter yesterday, please see below:

Refer to Andrew Bremner & M. Ulas 2011 paper in the International journal of number theory, pages 2018-2090, vol. 8, No 07, having title $ (x^a±y^b±z^c±w^d=0)$

The paper includes additional numerical solutions to equation (1) above:

$(x, y, z, w )$

$694, 945, 1308, 2414891825$

$42, 873, 3596, 46505412377$

$792, 3759, 5038, 138465240337$

$1515, 3262, 5160, 141747483853$

$2975, 4950, 7902, 508783710817$

$4410, 5463, 8270, 594854319097$

$5340, 6626, 9765, 987341285501$

$1689, 10528, 14886, 3498954949801$

$588, 8224, 26097, 17782152244433$

$834, 17094, 21373, 10966834991269$

$1182, 14644, 24597, 15209227541197$