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I tried to enter this question:

sage: var('x y')
(x, y)
sage: eq1=16^(x^2+y)+16^(y^2+x)-1
sage: solve(eq1,x,y)
[]

But it gives empty result []. The known result for this question is (x,y)=(-1/2,-1/2).

I tried to enter this question:

sage: var('x y')
(x, y)
sage: eq1=16^(x^2+y)+16^(y^2+x)-1
eq1=16^(x^2+y)+16^(y^2+x)==1
sage: solve(eq1,x,y)
[]

But it gives empty result []. The known result for this question is (x,y)=(-1/2,-1/2).