  I investigate the higher-derivative conformal theory which shows how the
Nambu-Goto and Polyakov strings can be told apart. Its energy-momentum tensor
is conserved, traceless but does not belong to the conformal family of the unit
operator. To implement conformal invariance in this case, I develop the new
technique that explicitly accounts for the quantum equation of motion and
results in singular products. I show that the conformal transformations
generated by such a nonprimary energy-momentum tensor form a Lie algebra with a
central extension which in the path-integral formalism gives a logarithmically
divergent contribution to the central charge. I demonstrate how the logarithmic
divergence is canceled in the string susceptibility and reproduce the
previously obtained deviation from KPZ-DDK at one loop.
