The problem $⊕2SAT$ is defined as a problem where we need to find the parity of the number of solutions of $2$-$CNF$ formulae and is known to be $\oplus P$ complete.
I introduce the following variant of $⊕2SAT$:
In this variant we put the restriction on the input of the $⊕2SAT$ problem the restriction is that every input must contain all clauses that contain a specific variable ,for concretness ,let's say we put the restriction on the input such that every input must contain all 2-literal clauses that contains variable (a) .
I want to ask if this variant of variant remains $\oplus P$ complete.
I think this variant is $\oplus P$ complete ,my reasons are that the output of this type of problem is restricted to either true or false thus ,I think even if we apply the above restriction the input should be able to express any $⊕2SAT$ without changing the output of the $⊕2SAT$ instance.