Balanced parenthesis, with backslash escaping
Superficial remarks

Whats a "bacelet", in isBaceletFind()?  Did you mean "bracelet"?  Even that is questionable English terminology, as I would take it to mean a piece of jewelry worn at the wrist.  Or maybe you meant "bracket"?

isPopable() should be isPoppable().

I normally don't like to be pedantic about brace styles, but the one you use for your if statements is really very unconventional, and certainly not easy to read.  Wikipedia lists 8 common brace styles, and yours is not any of them.  In fact, I'd say that it's the first time I've ever seen it in use.

Data structure

java.util.Stack is deprecated, because it has bad object-oriented design (it allows random access because it extends java.util.Vector), and because its methods are synchronized (which incurs unnecessary performance overhead).  Normally, the recommended replacement for Stack is ArrayDeque.  However, in this case, since each element is a char, I'd use a StringBuilder instead, so that you don't have to incur the char ↔︎ Character boxing/unboxing overhead.

Even if your code works, it's a bad practice to write:


private static final Stack&lt;Character&gt; STACK = new Stack&lt;&gt;();




static means that there is only one instance shared throughout the Java interpreter.  Having to write STACK.clear() at the beginning of isBalanced() is ugly.  It also unnecessarily introduces the possibility of multithreading problems, when isBalanced() should be a simple deterministic function with no threading issues.
The ALL_CAPS name suggests that it's a constant, which it isn't in nature, even if it's technically a final variable.


Instead, isBalanced() should instantiate the stack as a local variable, and explicitly pass it to any helper function that needs it.

Algorithm

I'm not a big fan of the BRACELETS constants, with the first half being the opening delimiters, with their counterparts in the second half.  I think it's more cryptic than it needs to be.  I'd just write a switch containing all of the characters of interest: it compiles to very clean and efficient bytecode, even simpler than String.indexOf().

When you encounter an opening delimiter, it would be smarter to push its counterpart onto the stack instead.  That way, you don't have to repeatedly ask "is this the right closing delimiter?" when looking at every subsequent character.

public class BalancedParenthesis {
    private static class CharStack {
        private StringBuilder sb = new StringBuilder();
        public boolean isEmpty() { return this.sb.length() == 0; }
        public void push(char c) { this.sb.append(c); }
        public char pop() {
            char last = this.sb.charAt(this.sb.length() - 1);
            this.sb.setLength(this.sb.length() - 1);
            return last;
        }
    }

    public static boolean isBalanced(String str) {
        CharStack stack = new CharStack();
        for (int pos = 0; pos &lt; str.length(); pos++) {
            char ch = str.charAt(pos);
            switch (ch) {
              // Backslash: ignore the next character
              case '\\': pos++; continue;

              // Opening delimiters
              case '(':  stack.push(')'); break;
              case '{':  stack.push('}'); break;
              case '[':  stack.push(']'); break;
              case '&lt;':  stack.push('&gt;'); break;

              // Closing delimiters
              case ')':
              case '}':
              case ']':
              case '&gt;':
                if (stack.isEmpty() || stack.pop() != ch) {
                    return false;
                }
            }
        }
        return stack.isEmpty();
    }
}