During the past days, I asked some questions in order to gain a clear understanding of the notion of "free algebras". I suppose that the question below is the most clear image of the concept I have in my mind:

Let $\mathcal{L}$ be an algebraic language. A negated identity in $\mathcal{L}$ is a formula of the form $$ \forall x_1 \ldots \forall x_n: p(x_1, \ldots, x_n)\neq q(x_1, \ldots, x_n), $$ where $p$ and $q$ are terms in $\mathcal{L}$. For an algebra $A$ of type $\mathcal{L}$ assume that $id^+(A)$ and $id^-(A)$ are the set of all identities and negated identities valied in $A$, respectively. As we know $$ Var(A)=Mod(id^+(A)) $$ is the variety generated by $A$ and during the questions

we tried to understand the relative free algebras in $Var(A)$. Now, suppose $$ Var^-(A)=Mod(id^-(A)). $$ It is easy to see that $Var^-(A)$ is $\{ S, P\}$-closed and so it is a pre-variety. Hence, for any set $X$, there exists a free algebra $F_{Var^-(A)}(X)$ in this pre-variety. Now, we ask:

How we can determine the structure of $F_{Var^-(A)}(X)$? I mean an answer like polynomial functions given by Anton Klaychko or the extendblity criterion given by Benjamin Steinberg in my previous question.

Can we characterize $Var^-(A)$ using class operators? I mean some thing like HSP.

P.S. To answer the question $F_{Var^-(A)}(X)=$?, one should determine the set of all identities with variables from $X$ which are logical consequence of the set $id^-(A)$. More precisely, suppose $$ R(X)=\{ (p, q):\ id^-(A)\vDash p=q\}. $$ Then we have $F_{Var^-(A)}(X)=T_{\mathcal{L}}(X)/R(X)$, where $T_{\mathcal{L}}(X)$ is the term algebra. So, to answer the first question, one should say that: which identities are logical consequences of given negated identities. For example, suppose $\mathcal{L}=(0,1, +, -, \times)$. Is it possible to determine all non-trivial identities which can be deduce from the negated identity $$ \forall x \forall y: x^2+y^2\neq-1? $$ Honestly, I have no even one example of non-trivial identity deducible from the above negated identity.