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<p>You are given a string <code>num</code>, representing a large integer. Return <em>the <strong>largest-valued odd</strong> integer (as a string) that is a <strong>non-empty substring</strong> of </em><code>num</code><em>, or an empty string </em><code>&quot;&quot;</code><em> if no odd integer exists</em>.</p>
<p>A <strong>substring</strong> is a contiguous sequence of characters within a string.</p>
<p>&nbsp;</p>
<p><strong>Example 1:</strong></p>
<pre>
<strong>Input:</strong> num = &quot;52&quot;
<strong>Output:</strong> &quot;5&quot;
<strong>Explanation:</strong> The only non-empty substrings are &quot;5&quot;, &quot;2&quot;, and &quot;52&quot;. &quot;5&quot; is the only odd number.
</pre>
<p><strong>Example 2:</strong></p>
<pre>
<strong>Input:</strong> num = &quot;4206&quot;
<strong>Output:</strong> &quot;&quot;
<strong>Explanation:</strong> There are no odd numbers in &quot;4206&quot;.
</pre>
<p><strong>Example 3:</strong></p>
<pre>
<strong>Input:</strong> num = &quot;35427&quot;
<strong>Output:</strong> &quot;35427&quot;
<strong>Explanation:</strong> &quot;35427&quot; is already an odd number.
</pre>
<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>
<ul>
<li><code>1 &lt;= num.length &lt;= 10<sup>5</sup></code></li>
<li><code>num</code> only consists of digits and does not contain any leading zeros.</li>
</ul>