Wolfram Language (Mathematica), 33 26 bytes

I just noticed that for no integer solution, the output is undefined instead of empty list; that allows to remove a few bytes.

x/.Solve[#==0,x,Integers]&

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Now if no integer solution exists, the function returns x.

Previously:

x/.Solve[#==0,x,Integers]/.x->{}&

Try it online!

Wolfram Language (Mathematica), 33 26 31 bytes

Fixed an error noted by Kelly Lowder in the comments.

x/.{}⋃Solve[#==0,x,Integers]&

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Previous incorrect solutions:

I just noticed that for no integer solution, the output is undefined instead of empty list; that allows to remove a few bytes.

x/.Solve[#==0,x,Integers]&

Try it online!

Now if no integer solution exists, the function returns x.

Previously:

x/.Solve[#==0,x,Integers]/.x->{}&

Try it online!

Wolfram Language (Mathematica), 33 bytes

x/.Solve[#==0,x,Integers]/.x->{}&

Try it online!

Wolfram Language (Mathematica), 33 26 bytes

I just noticed that for no integer solution, the output is undefined instead of empty list; that allows to remove a few bytes.

x/.Solve[#==0,x,Integers]&

Try it online!

Now if no integer solution exists, the function returns x.

Previously:

x/.Solve[#==0,x,Integers]/.x->{}&

Try it online!