Can a spiral be represented implicitly as $F(x)+G(y)+H(x+y) = 0$, where $F,G,H$ are continuous differentiable real-valued functions?
Definition: A spiral is a simple curve $C\subset \mathbb R^2$ with parametric representation $$ C_i = \Big\{ \Big( r(t) \sin\big(\phi(t)\big), r(t) \cos\big(\phi(t)\big) \Big), t\in \mathbb R_+ \Big\}, $$ where both $r$ and $\phi$ are continuous functions such that $r(0)=0$, and both $\phi(t)$ and $r(t)$ go to infinity as $t\to\infty$.
The implicit form $F(x)+G(y)=0$ is very restrictive, in the post Can a spiral be represented as $F(x)+G(y)=0$ it was argued that no spiral can be represented that way and more general case is discussed here: Can an $n$-armed spiral be represented as $F(x)+G(y)=0$?. The form is also very restrictive on closed curves that can be represented that way: Closed curves of the form $F(x)+G(y)=0$.
I'm interested whether adding "one degree of freedom" in the implicit formula by including the term $H(x+y)$ in it will allow for different types of curves to be representable that way.