The term *Lax pair* is fairly often used in a broader sense as a pair of linear equations (not necessarily of the form you mention, i.e., $L\phi=\lambda\phi$ and $\dot{\phi}=B\phi$) whose compatibility condition is the system under study. In fact, this notion is occasionally stretched even further, for example, in the theory of integrable dispersionless systems one encounters *nonlinear Lax pairs*, see e.g. [here][1] and references therein. 


  [1]: http://arxiv.org/abs/1401.2122