x =   ====>  


x(y^2-x) = 4 - x

xy^2 - x^2 = 4 - x

xy^2 = x^2 - x + 4

y^2 = 


y^2 = x - 1 + .    (1)


y is integer. So, y^2 is integer.  x is integer.  So, (x-1) is integer.


It implies that  "x" is the solution to the problem if and only if the value    is positive integer.  

It implies, in turn,  that "x" may have only these values: x= 1, 2 and/or 4.


Then from (1)  y^2 = 4, 3 and 1, respectively.


In order for  "y"  be integer,  y^2 can not be 3.


So, only one pair is the solution:  (x,y) = (1,2).