Satisfiability problems with restricted (not bounded) number of occurrences per variable Intro

It is known that SAT is hard even when restricted to, e.g., formulas with exactly 3 literals per clause and at most 4 occurrences per variable. On the other hand it is easy if there are exactly 3 literals per clause and at most 3 occurrences per variable. I am interested in other restrictions on the number of occurrences per variable.

For a set $S$ of positive integers, define $(k,S)$-SAT to be the satisfiability problem restricted to boolean formulas with exactly $k$ literals per clause, and the number of occurrences of each variable is in the set $S$. For example $(3,\{1,2,3,4\})$-SAT is hard, $(3,\{1,2,3\})$-SAT is easy.

Sub-questions

If $k=3$ and $S$ is the even numbers, then you can easily pad an instance of $3$-SAT to get an equivalent instance of $(3,S)$-SAT. Is there anything known more generally? Is it always hard if $S$ is infinite? Are there any easy cases for $k=3$ besides $S\subseteq \{1,2,3\}$?

Main Question

I am really interested in 1-in-3-SAT restricted to the case where each variable appears an even number of times - the standard reduction from 3-SAT introduces new variables some of which occur an odd number of times. My feeling is that the problem should be NP-hard. Is there a suitable reduction from SAT? Or from normal 1-in-3-SAT? Or perhaps some sort of dichotomy theorem I don't know about that can be applied?

Edit

The answer to my question as stated is that the problem is indeed hard, because you can simply copy all clauses of a 1-in-3-SAT instance. I now insist that the clauses should not be repeated (we can efficiently preprocess an instance of SAT so that there are no repeat clauses after all)
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 $I'$.