Boundedly isometric but not isometric spaces

The existence of a continuum of smooth complete (in intrinsic metrics) surfaces $\mathcal{M}_{\alpha}\subset\mathbb{R}^n$, $n\ge3$, is proved such that are homeomorphic to $\mathbb{R}^{n-1}$, any two of which are not isometric but possess the following property: every bounded domain on the first surface is isometrically embeddable into the second surface (and vice versa). Also, we prove the existence of $2^\mathfrak{c}$ subsets in the plane $\mathbb{R}^2$? (where $\mathfrak{c}$ is the cardinality of the continuum) each of which has diameter 1 and is embeddable into any other, with all the subsets pairwise nonhomeomorphic.