Fat point embeddings in affine space

We show that any fat point (local punctual scheme) has at most one embedding in the affine space up to analytic equivalence. If the algebra of functions of the fat point admits a non-trivial grading over the non-negative integers, we prove that it has at most one embedding up to algebraic equivalence. However, we give an example of a fat point having algebraically non-equivalent embeddings in the affine plane.