Satisfiability problems with restricted (not bounded) number of occurrences per variable
Theorem: The special case of 1-in-3-SAT where each variable 
  appears an even number of times is NP-hard.


Proof: Consider an instance $I$ of 1-in-3-SAT, and let $a_1,\ldots,a_n$ 
be an enumeration of the variables in $I$. 
Assume that variables $a_1,\ldots,a_m$ occur an odd number of times,
whereas $a_{m+1},\ldots,a_n$ occur an even number of times.
Without loss of generality $m$ is an even number (and otherwise, 
add a new dummy clause with three newly created dummy variables).

From instance $I$, we create a new instance $I'$ of 1-in-3-SAT 
in which every variable occurs an even number of times.
Furthermore, $I$ is satisfiable if and only if $I'$ is satisfiable:


Instance $I'$ contains all variables and all clauses of $I$.
For every variable $a_i$, instance $I'$ contains a (new)
corresponding variable $b_i$.
Furthermore, $I'$ contains a (new) variable $c$.
For every clause $a_i,a_j,a_k$ in $I$, instance $I'$ contains a new clause $\lnot b_i, \lnot b_j,\lnot b_k$.
For $i=1,\ldots,m$, instance $I'$ contains a new clause $a_i,b_i,c$.


It is easy to check that every variable in $I'$ occurs in an even
number of clauses. If $I'$ is satisfiable, then this induces a 
satisfying assignment for $I$.

Finally consider a satisfying assignment for $I$. 
Use truth values $b_i:=\lnot a_i$ for $1\le i\le n$ and $c:=$FALSE.
The resulting assignment satisfies all clauses in $I'$.