  This paper focuses on the analysis of stratified steady periodic water waves
that contain stagnation points. The initial step involves transforming the
free-boundary problem into a quasilinear pseudodifferential equation through a
conformal mapping technique, resulting in a periodic function of a single
variable. By utilizing the theorems developed by Crandall and Rabinowitz, we
establish the existence and formal stability of small-amplitude steady periodic
capillary-gravity water waves in the presence of stratified linear flows.
Notably, the stability of bifurcation solution curves is strongly influenced by
the stratified nature of the system. Additionally, as the Bernoulli's function
$\beta$ approaches critical values, we observe that the linearized problem
exhibits a two-dimensional kernel. Consequently, we apply a bifurcation theorem
due to Kielh\"{o}fer that incorporates multiple-dimensional kernels and
parameters, which enables us to establish the existence of two-mode water
waves. As far as we know, the two-mode water waves in stratified flow are first
constructed by us. Finally, we demonstrate the presence of internal stagnation
points within these waves.
