  In this note, we address the following question: Why certain nonassociative
algebra structures emerge in the regularity theory of elliptic type PDEs and
also in constructing nonclassical and singular solutions? The aim of the paper
is twofold. Firstly, to give a survey of diverse examples on nonregular
solutions to elliptic PDEs with emphasis on recent results on nonclassical
solutions to fully nonlinear equations. Secondly, to define an appropriate
algebraic formalism which makes the analytic part of the construction of
nonclassical solutions more transparent.
