Multivariate normal distribution, not full rank, probability density function

If X has a multivariate distribution of full rank and Y is the sum of the elements of X.

Then (X,Y) is a multivariate normal distribution but not of full rank, then as I understand you can not calculate a density function for it.

However would I be able to say the density function of (X,Y) is the same as the density function of X?

If true, then if given a multivariate distribution (say X) not of full rank. Can you obtain its density by looking at the density function of a subset of X which is of full rank?
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