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For solving the equation "xy=x", we can do:

sage: x,y=var('x,y')
sage: solve([x*y==x],[x,y])     
[x == 0]

But this is not the complete solution.

It suffices to add the trivial equation "0==0" for having the complete solution:

sage: solve([x*y==x,0==0],[x,y])
[[x == 0, y == r1], [x == r2, y == 1]]

Is it a bug ?

For solving the equation "xy=x", we can do:

sage: x,y=var('x,y')
sage: solve([x*y==x],[x,y])     
[x == 0]

But this is not the complete solution.

It suffices We need to add the trivial equation "0==0" for having the complete solution:

sage: solve([x*y==x,0==0],[x,y])
[[x == 0, y == r1], [x == r2, y == 1]]

Is it a bug ?