  We find a class of analytic solutions in open bosonic string field theory,
parametrized by the chiral copy of higher spin algebra in $AdS_3$. The
solutions are expressed in terms of the generating function for the products of
Bell polynomials in derivatives of bosonic space-time coordinates $X^m(z)$ of
the open string, which form is determined in this work. The products of these
polynomials form a natural operator algebra realizations of $W_\infty$
(area-preserving diffeomorphisms), enveloping algebra of SU(2) and higher spin
algebra in $AdS_3$. The class of SFT solutions found can, in turn, be
interpreted as the "enveloping of enveloping", or the enveloping of $AdS_3$
higher spin algebra.
  We also discuss the extensions of this class of solutions to superstring
theory and their relations to higher spin algebras in higher space-time
dimensions.
