It is an unnatural feature of the Standard Model that the
difference between the baryon number $B$ and the lepton number $L$
is conserved, which implies an exact, global symmetry.
We promote it to a naturally exact, local symmetry by coupling
the difference $B-L$ to an Abelian gauge field.
Gauge anomalies are cancelled by adding right-handed neutrinos
$\nu_{R}$. They are not sterile in this case, which
forbids the Majorana term, but we arrange for a
$\nu_{R}$-mass by adding a Higgs-type 1-component complex scalar
field. Thus we arrive at a well-motivated, modest
extension of the Standard Model (SM).

We investigate the field equations in the corresponding
gauge-Higgs sector, involving both Higgs fields and the non-standard
U(1) gauge field. This leads to a set of coupled, non-linear
differential equations, which we solve numerically. We use
cylindrical coordinates, thus focusing on
the structures of cosmic strings. For a large variety of parameters,
we identify the string profiles and energy densities.
These profiles depend on the winding number of each of the Higgs
fields. For our parameters, the characteristic string radii are
below $10^{-3}~{\rm fm}$, and the string tension is not ruled out
by observations of the Cosmic Microwave Background or gravitational
waves. As an amazing peculiarity, we discover - for high winding
numbers - ``overshooting'' and ``co-axial'' profile functions. In
the latter case, the profile of the standard Higgs field changes
its sign near the core of the cosmic string.