= Solution
Write
$$
S(x):=\forall y(E(x,y)\to y=x),\qquad R(x):=\exists y(E(x,y)\land y\ne x).
$$
These are <first-order formulas> in the original language, not new predicate symbols. An axiomatization consists of the <equivalence relation> axioms for $E$, the bound
$$
\forall x\forall y\forall z\bigl((E(x,y)\land E(x,z))\to(y=z\lor y=x\lor z=x)\bigr),
$$
and, for every $n\geq1$, the two sentences
$$
\exists x_1\cdots x_n\left(\bigwedge_iS(x_i)\land\bigwedge_{i<j}x_i\ne x_j\right),
$$
$$
\exists x_1\cdots x_n\left(\bigwedge_iR(x_i)\land\bigwedge_{i<j}\neg E(x_i,x_j)\right).
$$
The first scheme supplies infinitely many singleton classes. The second supplies infinitely many distinct non-singleton classes; the size bound makes these doubletons.
Every countable <first-order model> has countably infinitely many classes of each size. A bijection between singleton classes, and another between doubleton classes followed by a bijection inside each matched pair, gives an <isomorphism> between any two countable <first-order models>. To infer completeness, suppose a sentence had different truth values in two arbitrary <first-order models>. The <Downward Lowenheim-Skolem theorem> supplies countable <elementary substructures> of those <first-order models>, which would still disagree on the sentence, contradicting the <isomorphism> just constructed. Thus all <first-order models> satisfy the same sentences. Since the original <first-order structure> satisfies the axioms, \b[these axioms form a complete axiomatization of its <first-order theory>]. This is the <theory of infinitely many singleton and doubleton equivalence classes>.
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