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When the displayed equality holds, the two nonzero eigenvalues of the three-state cyclic conversion model coincide at $\lambda =-(a+b+c)/2$. The positive cyclic rates force each eigenspace to have dimension at most one: if its $A$ coordinate vanished, the first eigenvector equation would force $C=0$, and then the third would force $B=0$. Thus the repeated eigenvalue has only one eigenvector direction and the matrix is not diagonalizable. A length-two Jordan chain produces both $e^{\lambda t}$ and $t{e}^{\lambda t}$ transients. The rates $(a,b,c)=(1,4,1)$ give an explicit example with $\lambda =-3$.

Reading the coefficients in the ordered concentration vector gives the linear system of ordinary differential equationsIts columns sum to zero, so $d(A+B+C)/dt=0$. This three-state cyclic conversion model conserves total concentration.