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When does this not exist between original parties when analyzing the running of a covenant?

This will depend on whether your professor teaches the modern and classic rule or just the modern rule.

Classic Rule- both parties must maintain an interest in the land i.e. an easement, profit, Reversion, Right of Reverter, Right of Re-entry, mortgage, etc.

Modern Rule- the covenant just has to been in the deed at the time of the conveyance

So if neighbors make an agreement to both not take a certain action on their land, the burden would not run depending on the approach regarding horizontal privity??

Exactly, in your case you would need the parties to both maintain an interest in the burdened properties, if they did not they would have no vertical privity.

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Sorry I meant Horizontal