The following two are very basic axioms put upon the successor function $S$: \begin{align} \neg Sx&=0\tag{1}\\\ Sx=Sy&\rightarrow x=y.\tag{2} \end{align}

Let $PA^{(-1)}$ be the subtheory of $PA$ (first-order Peano Arithmetic) which has all other axioms except for (1), similarly $PA^{(-2)}$ let be the subtheory without (2). It is rather easy, but nevertheless interesting, result that both this theories have finite models, $PA^{(-1)}$ even has the degenerate one-element model.

My question is: has any research been made towards characterization of class of models of the theories above? If yes, could please someone provide me with suitable information? I am particularly interested in finite models.

Consider first-order theory (with identity) of Peano Artithmetic built in the language $\{S,+,\times,0\}$ and with the following set of axioms: \begin{align} \neg Sx&=0\tag{1}\\\ Sx=Sy&\rightarrow x=y\tag{2}\\\ x+0&=x\tag{3}\\\ x+S(y)&=S(x+y)\tag{4}\\\ x\times 0&=0\tag{5}\\\ x\times S(y)&=(x\times y)+x\tag{6} \end{align} plus the full induction schema.

Let $PA^{(-1)}$ be the subtheory of $PA$ (first-order Peano Arithmetic) which has all other axioms except for (1), similarly $PA^{(-2)}$ let be the subtheory without (2). It is rather easy, but nevertheless interesting, result that both this theories have finite models, $PA^{(-1)}$ even has the degenerate one-element model.

My question is: has any research been made towards characterization of class of models of the theories above? If yes, could please someone provide me with suitable information? I am particularly interested in finite models.

EDIT: Following J.D. Hamkins advice I explicitly stated the language and the axiomatization I am interested in.