Complexity of this variant of $⊕2SAT$?

Question

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.

Answer 1

This problem can be solved easily: trivially, the number of solutions is 0, so the parity is even.

Consider any other variable $b$. Then because the formula $\varphi$ must contain all clauses that contain the variable $a$, $\varphi$ must have the form

$$\varphi = (a \lor b) \land (a \lor \neg b) \land (\neg a \lor b) \land (\neg a \lor \neg b) \land \cdots$$

It is easy to see that no assignment can satisfy this formula. As such, it is very unlikely that the problem is $\oplus P$ complete.

The reasoning you give makes no sense to me. There are plenty of problems whose output are true or false but are not $\oplus P$ complete.