  We find the connection between relativistic and non-relativistic string vacua
in AdS$_5 \times$S$^5$ in terms of a free parameter $c$ flow. First, we show
that the famous relativistic BMN vacuum flows in the large $c$ parameter to an
unphysical solution of the non-relativistic theory. Then, we consider the
simplest non-relativistic vacuum, found in arXiv:2109.13240 (called BMN-like),
and we identify its relativistic origin, namely a non-compact version of the
folded string with zero spin, ignored in the past due to its infinite energy.
We show that, once the critical closed B-field required by the non-relativistic
limit is included, the total energy of such relativistic solution is finite,
and in the large $c$ parameter it precisely matches the one of the BMN-like
string. We also analyse the case with spin in the transverse AdS directions.
