Let $f\in\mathcal{C}^1(\mathbb{R}^d,\mathbb{R}^d)$. I would like to construct a differentiable function $\Phi:\mathbb{R}^d\times\mathbb{R}$ such that $\Phi(\cdot,0)=f$ and $\det(\nabla_x\Phi)=0\implies \frac{\partial}{\partial \theta}(\nabla \Phi_x)\ne0$
In the case $d=1$, $\Phi(x,\theta)=f(x)+\theta x$ works
In the case $d=2$ I was able to show that it isn't possible for $\Phi(x,\theta)=f(x)+A(\theta)B(x)$ for any $A:\mathbb{R}\to\mathbb{R}^{2\times 2}$ and $B:\mathbb{R}^2\to\mathbb{R}^2$