Question: How make System from ODE when contain singularity? then finding `conserved quantity` ?

in here i want the system which when i found the system it contain a singularity but by some changing of system we can remove it, i know how he did that but i can't do it by coding, also after removing which i did by hand , there is another problem which equalibruiom points is to long how i can make it be shorter becuase contain two function i don't know how do that can i change the coiefecient of any function by another letter? also for phase portrait i need `conserved quantity` which last code is not work for finding this kind of system ? and how decide about jacobian if the `conserved quantity` is know by that after determine the points it will be clear on plots, i write the code untill finding the `conserved quantity` becuase code not run i didn't find jacobian and phase portrait but i will put the code of jacobian and phase portrait too! 

bi-1.mw