Here is an infinite family of solutions resulting from setting $x=A z$ for integer $A$.

For example set $x=2z$ and get

$$g(y,z)=-(y^2 - 8*z^2 - z)*z$$

The quadratic factor is conic and Wolfram Alpha gives infinitely many integer solutions in terms of powers of square root of two, e.g: $f(2*36,102,36)=0$

Potential attack might be to try rational $A$ and then find integral points on a conic with rational coefficients.

There exists parametrization by rational functions since your equation is a rational surface:

$X1=-1/2*s^2*t/(2*t^3 - s),Y1=-1/2*s^2/(2*t^3 - s),Z1=1/2*s^3*t^3/(4*t^6 - 4*s*t^3 + s^2)$

Here is an infinite family of solutions resulting from setting $x=A z$ for integer $A$.

For example set $x=2z$ and get

$$g(y,z)=-(y^2 - 8*z^2 - z)*z$$

The quadratic factor is conic and Wolfram Alpha gives infinitely many integer solutions in terms of powers of square root of two, e.g: $f(2*36,102,36)=0$

Potential attack might be to try rational $A$ and then find integral points on a conic with rational coefficients.

There exists parametrization of the rational solutions since your equation is a rational surface:

$X1=-1/2*s^2*t/(2*t^3 - s),Y1=-1/2*s^2/(2*t^3 - s),Z1=1/2*s^3*t^3/(4*t^6 - 4*s*t^3 + s^2)$

Here is an infinite family of solutions resulting from setting $x=A z$ for integer $A$.

For example set $x=2z$ and get

$$g(y,z)=-(y^2 - 8*z^2 - z)*z$$

The quadratic factor is conic and Wolfram Alpha gives infinitely many integer solutions in terms of powers of square root of two, e.g: $f(2*36,102,36)=0$

Potential attack might be to try rational $A$ and then find integral points on a conic with rational coefficients.

Here is an infinite family of solutions resulting from setting $x=A z$ for integer $A$.

For example set $x=2z$ and get

$$g(y,z)=-(y^2 - 8*z^2 - z)*z$$

The quadratic factor is conic and Wolfram Alpha gives infinitely many integer solutions in terms of powers of square root of two, e.g: $f(2*36,102,36)=0$

Potential attack might be to try rational $A$ and then find integral points on a conic with rational coefficients.

There exists parametrization by rational functions since your equation is a rational surface:

$X1=-1/2*s^2*t/(2*t^3 - s),Y1=-1/2*s^2/(2*t^3 - s),Z1=1/2*s^3*t^3/(4*t^6 - 4*s*t^3 + s^2)$