How to determine system order with degeneracy?
The state variable of an inductor is generally chosen to be the current, because it can not be changed abruptly, it is an adequate variable to be differentiated. The inductor voltage on the other hand, can change abruptly. 

For the capacitor it is the opposite, voltage is chosen as the state variable.

A practical rule of thumb to find the system order can be: find the number of independent inductor currents and capacitor voltages.

In the circuit from your question the inductor current is not at independent variable as it is fixed by the current source, hence the order 1. If you change this to a voltage source the inductor current is independent from it, so it is the capacitor voltage, hence the order 2.

If you want a more theoretical way to approach at it you can write down the differential equations that describe the circuit. Then find its rank like explained here:
https://en.wikipedia.org/wiki/Rank_(linear_algebra)

If you need some background on that, you can go to a Dynamic Systems textbook or here:
https://en.wikipedia.org/wiki/System_of_linear_equations