Discrete and nondiscrete isometric deformations of surfaces in $\mathbb R^3$

We prove existence of closed infinitely differentiable surfaces $M$ of $\mathbb R^3$ each of which can be included in some family $F$ of isometric pairwise noncongruent infinitely differentiable surfaces which is uniformly as close as we want to $M$. We also prove that $F$ can be more than countable.