1. In <a href="http://www.ams.org/mathscinet-getitem?mr=232018">Artin1968</a> he considers $\underline{analytic}$ equations, but over the ring $R=k\{x_1,..,x_n\}$. In <a href="http://www.ams.org/mathscinet-getitem?mr=268188">Artin1969</a> he works with $R=k\{x_1,..,x_n\}/I$, not necessarily regular, but considers $\underline{polynomial}$ equations.
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Is there some version like this: "Let $R$ be a local Noetherian Henselian ring(not necessarily regular), over a normed field. Given an arbitrary (possibly countable) system of analytic equations over $R$, with a solution over the completion of $R$, there exists also a solution in $R$, sufficiently close to the formal solution" ??

2. What is known for non-Noetherian rings? e.g. for $C^\infty$, $C^r$?
<br> (Actually, for $C^\infty$ I learned about one approximation theorem, unpublished in the old  USSR times..)