  Motivated by recent experimental observation of spin-orbit coupling in carbon
nanotube quantum dots [F. Kuemmeth \textsl{et al.}, Nature (London) {\bf 452},
448 (2008)], we investigate in detail its influence on the Kondo effect. The
spin-orbit coupling intrinsically lifts out the fourfold degeneracy of a single
electron in the dot, thereby breaking the SU(4) symmetry and splitting the
Kondo resonance even at zero magnetic field. When the field is applied, the
Kondo resonance further splits and exhibits fine multipeak structures resulting
from the interplay of spin-orbit coupling and Zeeman effect. A microscopic
cotunneling process for each peak can be uniquely identified. Finally, a purely
orbital Kondo effect in the two-electron regime is also obtained.
