Deterministic pushdown automaton for a given language
Let us start with an automaton for ${2^rc^r}$ for the alphabet $\{2, c\}$. This is quite a standard example. As long as you are reading $2$ add a letter to the stack until you read the first $c$. Then for each $c$ pop a letter from the stack. The word and the stack should finish at the same time.

Now using this automaton we should be able to build one for you language, where the string of $c$s should turn to any concatenation of a staring of $a$s and a string of $b$s and the string of $2$s turns into a concatenation of a string of ones and a string of twos. This means for each zero you read push a letter to the stack until you read the first one. Then for each one push a letter to the stack until you read the first $a$ then for each $a$ pop a letter until you find the first $b$ and then pop for each $b$. The stack should turn empty upon reading the last letter in the world. Note that the DPA should rejects the string if at any point it reads an unexpected letter (a $0$ after an $a$ for example).

Note that one of the substrings might be empty, so you should consider the case of starting with $a$s directly after zeroes for example.