If h {\mathbf {h}} and k {\mathbf {k}} are vector 1 1 -fotms, the vanishing of the concomitant [ h , k ] [{\mathbf {h}},\;{\mathbf {k}}] is an integrability condition fot certain problems on manifolds. In the case that h = k {\mathbf {h}} = {\mathbf {k}} the vanishing of the Nijenhuis tensor [ h , h ] [{\mathbf {h}},\;{\mathbf {h}}] implies d ( tr ⁡ h ) d(\operatorname {tr} {\mathbf {h}}) is a conservation law for h {\mathbf {h}} , provided that tr ⁡ h \operatorname {tr} {\mathbf {h}} is not constant. When the trace of h {\mathbf {h}} is constant, a conservation law for h {\mathbf {h}} exists if one can find a vector 1 1 -form k {\mathbf {k}} with nonconstant trace such that [ h , k ] = 0 [{\mathbf {h}},\;{\mathbf {k}}] = 0 .