An interesting example of a compact non-C-analytic real subvariety of $R^3$

The purpose of this short note is to provide an interesting new example exhibiting some of the pathological properties of real-analytic subvarieties. We construct a compact irreducible real-analytic subvariety $S$ of $R^3$ of pure dimension two such that 1) the only a real-analytic function is defined in a neighbourhood of $S$ and vanishing on $S$ is the zero function, 2) the singular set of $S$ is not a subvariety of $S$, nor is it contained in any one-dimensional subvariety of $S$, 3) the variety $S$ contains a proper subvariety of dimension two. The example shows how a badly behaved part of a subvariety can be hidden via a second well behaved component to create a subvariety of a larger set.