Can we find integers x and y such that f,g,h are strictely positive integers

Let $a>2$ and $b>2$ two strictely positive integers. Let us consider the following quantities:

$$f={\displaystyle \frac{xy+ay+{a}^{2}}{by}}$$

$$g=a(y+a){\displaystyle \frac{xy+ay+{a}^{2}}{b{y}^{2}x}}$$

$$h={\displaystyle \frac{y+a}{b}}$$

My question is:

Can we find integers x and y (not necessarly positive) such that f,g,h are strictely positive integers. Or at lest how one can proves that they are exist.

Let $a>2$ and $b>2$ two strictely positive integers. Let us consider the following quantities:

$$f={\displaystyle \frac{xy+ay+{a}^{2}}{by}}$$

$$g=a(y+a){\displaystyle \frac{xy+ay+{a}^{2}}{b{y}^{2}x}}$$

$$h={\displaystyle \frac{y+a}{b}}$$

My question is:

Can we find integers x and y (not necessarly positive) such that f,g,h are strictely positive integers. Or at lest how one can proves that they are exist.