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A general method, not necessarily best for this particular equation, is to split into three cases by writing $x=3u+v$ with $v\in\{0,1,2\}$. Then rewrite your equation as $$ A(7^u)^3 + 2 = y^2\quad\text{with $A\in\{1,7,49\}$.} $$ So any solution to your equation gives an integer solution to one of the three equations $$ w^3+2=y^2,\quad 7w^3+2=y^2,\quad49w^3+2=y^2.$$ There are then standard methods for finding integer points on these genus $1$ curves.