What is the mechanism for oxidation of ketones by the Popoff's rule?
I am uncertain about the "Popoff's" rule you mention. There are two reactions that can oxidize ketones, and one seems to follow the behavior you are suggesting, but does not form a carboxylic acid. The other reaction does form carboxylic acids, but is more complex.

The Baeyer-Villager oxidation is an oxidation of ketones to esters using a peracid in the presence of a mild base:

$$\ce{CH3COCH3 + RCO3H-&gt;[\ce{Na2HPO4}] CH3CO-OCH3 + RCOOH}$$ 

The mechanism involves the fragmentation of one of the $\ce{C-CO}$ bonds. Unsymmetrical ketones fragment in a predictable pattern, but not always that the carbonyl remains with the smaller group. The fragment that would be a more stable carbocation (even though carbocations are not formed in this reaction) is the one to move. The migratory aptitude is tertiary alkyl > cyclohexyl > secondary alkyl, aryl > H > primary alkyl > methyl . For exampl, with 2-methyl-3-octanone, the isopropyl group moves because it is attached via a $2^\circ$ carbon:

$$\ce{(CH3)2CHCOCH2CH2CH2CH2CH3 + RCO-O-OH -&gt;[\ce{Na2HPO4}] (CH3)2CH{\bf O}COCH2CH2CH2CH2CH3 + RCOOH}$$

Nitric acid chews ketones apart into two carboxylic acids. Concentrated $\ce{KMnO4}$ in acid also does this. These reactions are a little bit harder to find information on, since they tend to be considered uncontrollable. 

The reaction goes through a series of oxidations from ketone to $\alpha$-hydroxyketone to cleaved acids through several enol intermediates. 

$$\ce{RCH2COCH2R &lt;=&gt;[\ce{HNO3}] RCH=C(OH)CH2R -&gt;[\ce{HNO3}] RCH(OH)COCH2R}$$
$$\ce{RCH(OH)COCH2R &lt;=&gt;[\ce{HNO3}] RC(OH)=C(OH)CH2R -&gt;[\ce{HNO3}] RCO}$$

If the ketone is unsymmetrical, there is no guarantee that is will cleave predictably. For example, 2-butanone could cleave into propanoic acid and carbon dioxide or two equivalents of acetic acid.

$$\ce{CH3CH2COCH3 -&gt;[\ce{HNO3}] n(CH3CH2COOH + CO2) + (1-n)(2CH3COOH)}$$

A balanced equation for the formation of acetic acid would look like:
$$\ce{CH3CH2COCH3 + 3NO3^- +  -&gt; 2 CH3CO2H + 3NO2^-}$$

A balanced equation for the formation of propanoic acid and carbon dioxide would look like:
$$\ce{CH3CH2COCH3 + 4NO3^- +  -&gt; CH3CH2COOH + CO2 + 4NO2^- + H2O}$$

The permanganate reactions are tougher to balance, since permanganate is a three electron oxidant.



Update:

When I wrote this answer, I had never heard of Popoff's rule. No textbook I own mentions this rule, and a Google search about it brings up this question as the top hit (and similar questions at other sites like ask.yahoo.com as the other hits). I now know that Aleksandr Popov published a paper in Liebigs Annalen in 1872 describing a variation of the reaction that would become the Baeyer-Village oxidation. This article is behind a paywall for me, and the first page preview confirms I would not be able to make much of it. I only know a small amount of German and the PDF sadly looks like a low quality copy of a copy of a copy.

However, from the title "Die Oxydation der Ketone als Mittel zur Bestimmung der Constitution der fetten Säuren und der Alkohole" I can parse what the paper was about. Roughly this paper was about "The oxidization of ketones - a means for determining the constitution of the fatty acids and alcohols". Thus, his method involved the oxidation to the ester and hydrolysis of said ester in one step. I don't know what reagents Popov used, but I'm pretty sure it was not a peroxyacid (since this reagent will not hydrolyze the ester). I have previously answered a question about migratory aptitude in Baeyer-Villager reactions. Since I cannot read Popov's paper, nor can I find any authoritative resource on his rule (on- or offline), I have to assume that the known migratory aptitude for the Baeyer-Village reaction is the same as Popov's rule.