Why are the integral points on elliptic curve preserved under this substitution?

Question

I have the following elliptic curve:

$$3b^2-b=a^3+a^2$$

When I subsitute $\left(a,b\right)=\left(\frac{X-1}{3},\frac{Y+2}{9}\right)$, I get the folloing minimal Weierstrass equation:

$$Y^2+Y=X^3-3X+4$$

Question: how do I now know that I have all the integral points for the elliptic curve with $a$ and $b$? My prof. told me that under my substitution the integral points are preserved but why is that the case?

Answer 1

If $a$ and $b$ are integers, then $X = 3a+1$ and $Y = 9b-2$ are integers. If you know the points with integer $(X, Y)$-coordinates, you know the points with integer $(a, b)$-coordinates.