CK Vul is the remnant of an energetic eruption known as Nova 1670 that is thought to be caused by a stellar merger. The remnant is composed of (1) a large hourglass nebula of recombining gas (of 71arcsec size), very similar to some classical planetary and pre-planetary nebulae (PPNe), and (2) of a much smaller and cooler inner remnant prominent in millimeter-wave emission from molecules. We investigate the three-dimensional spatio-kinematic structure of both components. The analysis of the hourglass structure yields a revised distance to the object of >2.6 kpc, at least 3.7 times greater than so far assumed. At this distance, the stellar remnant has a bolometric luminosity >12 L$_{odot}$ and is surrounded by molecular material of total mass >0.8 M$_{odot}$ (the latter value has a large systematic uncertainty). We also analyzed the architecture of the inner molecular nebula using ALMA observations of rotational emission lines obtained at subarcsecond resolution. We find that the distribution of neutral and ionized gas in the lobes can be reproduced by several nested and incomplete shells or jets with different velocity fields and varying orientations. The analysis indicates that the molecular remnant was created in several ejection episodes, possibly involving an interacting binary system. We calculated the linear momentum ($approx$10$^{40}$ g cm s$^{-1}$) and kinetic energy ($approx$10$^{47}$ erg) of the CK Vul outflows and find them within the limits typical for classical PPNe. Given the similarities of the CK Vul outflows to PPNe, we suggest there may CK Vul analogs among wrongly classified PPNe with low intrinsic luminosities, especially among PPNe with post-red-giant-branch central stars.
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