Does a certain "Badecker reaction" actually exist?
The reaction in question is actually called the Boedeker reaction. How the book came from Boedeker to Badeker I do not know, but maybe there was one instance of Boedeker being written Bödeker which was later misparsed by a human or a robot and mistook for an a. (Replacing oe for ö is a common mistake with German names since the latter is typically used in orthography but names don’t adhere to orthography.)

The reaction is well-known and often used as a qualitative analysis for sulphite ions in solution, much like the same nitroprusside ion can also be used to detect sulphide ions. In chemical formulae, it is:

$$\ce{[Fe(CN)5(NO)]^2- + SO3^2- &lt;=&gt; [Fe(CN)5(NOSO3)]^4-}$$

The reaction in itself is also rather interesting since for over 100 years the structure of the addition complex was unknown. Bottomley et al. were finally able to obtain crystal structures of the probably structurally similar ruthenium complex $\ce{[RuCl(bpy)2(NOSO3)]}$.[1] These show that the addition product is — very much like in the sulphide case — formed by nucleophilic attack of the sulphite sulphur’s lone pair to nitrosyl nitrogen creating a nitrosylsulphite anion $\ce{O3SNO-}$. Thus, the complex formed should be named pentacyanidonitrosylsulphito-&#x3ba;N-ferrate(II).



Reference:
[1]: F. Bottomley, W. V. F. Brooks, D. E. Paéz, P. S. White, M. Mukida, J Chem. Soc., Dalton Trans. 1983, 2465. DOI: 10.1039/DT9830002465.