Recently T. Hou and G. Lou suggested a blow-up scenario for axi-symmetric incompressible 3d Euler equations, as well as a 1d model capturing certain features of the situation.  The model is related to equations previously introduced by Constantin-Lax-Majda and De Gregorio. We will discuss some rigorous results for the model. Joint work with K. Choi, T. Hou, A. Kiselev, G. Luo and Y. Yao.