  This paper answers some questions about conjugate points along the geodesics
corresponding to steady 2D Euler flows, posed by a paper of
Drivas-Misiolek-Shi-Yoneda. We present a new sufficient criterion for the
existence of conjugate points, which improves on the criterion of Misiolek. It
applies in any rotational cell of a steady 2D Euler flow, and in case of
rotational symmetry it captures all known conjugate points. We give a general
construction of the surfaces that admit a steady fluid with given area form,
velocity profile, and vorticity profile, and from this we show how to detect
conjugate points in a single rotational cell of a steady flow. When the
velocity profile has a local extremum, the criterion becomes particularly
simple. Several examples are provided, and in an appendix we use the
Misio{\l}ek criterion to give some new examples of conjugate points along
Kolmogorov flows on the torus.
