At first glance, I would say that it would be better to make the common code as a public static method in a public class. This will make the method useful to any class just by using
UtilityClassName.methodName();
This is better then making it a concrete method in an abstract super-class because then you will always need to extend this super-class in all the classes where you want to use this one single method.
But now, as you said that the method's behavior depends on some variables. Now, if it depends on the instance variables of different classes, then better add this method in an interface and let all your classes implement this interface and have their own implementation of the same.
But again if these variables are constant values, then have these constant values in an interface. Implement these interface in your utility class. And again make it a static method in that utility class which will directly use these constants.
For e.g. Consider foll. common code of returning area of a circle.
public interface TwoDimensional{
double PI = 3.14;
}
public class MyUtility implements TwoDimensional{
public static double getCircleArea(double radius){
return PI*radius*radius;
}
}
Here, you can see that method getCircleArea() depends on the radius which will be different for different classes but still I can pass this value to the static method of myUtility class.