Analysis: Using the undetermined coefficient method, according to the coordinates of all known points given in the known conditions, the equation is set as a general formula, and then according to the fact that the circle passes through two points A (4 4,2) and B (- 1, 3) and the sum of the four intercepts on the two coordinate axes is 2, the equation (group) is constructed and solved to get the answer.

Solution: Let the equation of the circle be x? +y？ +Dx+Ey+F=0。

Let y=0 and get x? +Dx+F=0，

The sum of the intercepts of a circle on the x axis is x 1+x2= -D,

Let x=0 to get y? +Ey+F=0，

The sum of the intercepts of a circle on the y axis is y 1+y2= -E,

Let the topic x1+x2+y1+y2 =-(d+e) = 2.

∴D+E= -2 ①

A (4 4,2) and B (- 1, 3) are on the circle,

∴ 16+4+4d+2e+f=0②

1+9-D+3E+F=0，③

D= -2，E=0，F F= - 12。 From the solution of ① ② ③.

So, the equation for finding a circle is: x? +y？ -2x- 12=0。

So the answer is: x? +y？ -2x- 12=0 .