How do I solve this gate with constant voltage at both ends?
There is current flow. Consider the picture on the RHS: the 5 branches are in parallel, driven by a given \$v_i\$ voltage (say, \$v_1\$). Then you can say the \$n-th\$ from the common node to ground will be:
$$i_n = \frac{v_i-L_{M_n}}{M(x)}$$
and the total current will be: \$i =\sum_{n=1}^4i_n+(v_i-L_R)/R\$. Hence you can see each branch will have a current depending on the value of the generator, whose values depend on the driving voltages and the constants you mentioned, and also on the value of the MR for a given \$x\$ value.
Then you can use MATLAB or Octave to see how it changes for various configurations. But the calculations are straightforward.