  New string models in D=4 space-time extended by tensor central charge
coordinates $z_{mn}$ are constructed. We use the $z_{mn}$ coordinates to
generate string tension using a minimally extended string action linear in
$dz^{mn}$. It is shown that the presence of $z_{mn}$ lifts the light-like
character of the tensionless string worldsheet and the degeneracy of its
induced metric. We analyse the equations of motion and find a solution of the
string equations in the generalized D=(4+6)-dimensional space $X^{\cal
M}=(x^{m},z^{mn})$ with $z_{mn}$ describing a spin wave process. A
supersymmetric version of the proposed model is formulated.
