Consider the function $$f(a,b,c,d):=\frac{\left(a^*\right)^2b^2-\left(b^*\right)^2a^2+\left(c^*\right)^2d^2-\left(d^*\right)^2c^2}{a^*a+c^*c}$$
With complex parameters $a,b,c$ and $d$
Now find any explicit $a(t), b(t), c(t)$ and $d(t)$, with $t$ is a arbitrary real number, such that $$f(t):=f\left(a(t),b(t),c(t),d(t)\right)=\text{const}$$
is constant with respect to $t$. That means $\frac{df}{dt}=0$ and at least one of them is not constant.
Finally I want to have a 8 dimensional "surjective" parametrization $a(p_1,...,p_8),b(p_1,...,p_8),c(p_1,...,p_8)$ and $d(p_1,...,p_8)$ such that $$f(p_1,...,p_8)=f(\vec p):=f\left(a(\vec p),b(\vec p),c(\vec p),d(\vec p)\right)$$ with $\frac{df}{dp_1}=...=\frac{df}{dp_7}=0$ but $\frac{df}{dp_8}\neq 0$.
The $p_i$ might be real. Instead of that 4 complex $p_i$ would also be fine.
As an example, consider $d(x_1,...,x_8):=\sum_{i=1}^8 x_i^2$. For this, that could be done using the polar coordinates. With seven angles, which let $g$ invariant and one radius which does not.
If I understand your extended question correctly (and the "polar coords" example helps here), I can propose the following:
$$ f(x_1, \ldots, x_8) = x_8 $$ This has seven partials that are all zero, and an eighth that is not. The function $f$ is invariant under seven independent translations, but not invariant under the 8th.
What I don't understand is how this is related to the first half of the problem, however, where there seems to be a curve in 4-space defined by $a(t), \ldots, d(t)$.