Every action $ \beta $ of $ \mathbb{G}_{a} $$ \mathbb{G}_a $ on a variety $ \operatorname{Spec}(A) $ over a field of characteristic zero is obtained from a locally nilpotent derivation $ \delta $ via $ f(t_{0} \ast x) = \sum_{j=0}^{\infty} (\delta^{j}(f)(x)t_{0}^{j})/j! $$ f(t_0 \ast x) = \sum_{j=0}^\infty (\delta^j(f)(x)t_0^j)/j! $. In this case, the existence of a local slice is equivalent to the existence of $ g,h \in A $ such that $ \delta(g)=h $ and $ h \in A^{\mathbb{G}_{a}} $$ h \in A^{\mathbb{G}_a} $.
Such a pair of polynomials always exists in characteristic zero. However, suppose that we now move to the category of formal schemes and possibly non-algebraic actions $ \beta $ of $ \widehat{\mathbb{G}_{a}} $$ \widehat{\mathbb{G}}_a $ on affine formal schemes $ \operatorname{Spf}(\widehat{A}) $. In this case, the action $ \beta $ is obtained from a derivation $ \delta \in \Omega_{\widehat{A}/k} $ via $ f(t_{0} \ast x) = \sum_{j=0}^{\infty} (\delta^{j}(f)(x)t_{0}^{j})/j! $$ f(t_0 \ast x) = \sum_{j=0}^\infty (\delta^j(f)(x)t_0^j)/j! $. Note that the derivation $ \delta $ may not be locally nilpotent so the co-action $ \beta^{\sharp}: \widehat{A} \to \widehat{A}[[t]] $. Also the action may not be algebraic. Does there exist a pair of $ g,h \in \widehat{A} $ such that $ \delta(g)=h $ and $ h \in \widehat{A}^{\widehat{\mathbb{G}_{a}}} $$ h \in \widehat{A}^{\widehat{\mathbb{G}}_a} $?
