Create a program in Java that will accept two integers. If the sum of the integers is odd, display the higher integer, else, display the lower integer.

Sure, here is a simple Java program that fulfills your requirements:

import java.util.Scanner;

public class Main {
    public static void main(String[] args) {
        Scanner scanner = new Scanner(System.in);

        System.out.println("Enter the first integer:");
        int num1 = scanner.nextInt();

        System.out.println("Enter the second integer:");
        int num2 = scanner.nextInt();

        int sum = num1 + num2;

        if (sum % 2 != 0) {
            System.out.println("The sum is odd. The higher integer is: " + Math.max(num1, num2));
        } else {
            System.out.println("The sum is even. The lower integer is: " + Math.min(num1, num2));
        }
    }
}

This program first asks the user to input two integers. It then calculates the sum of these integers. If the sum is odd, it displays the higher of the two integers. If the sum is even, it displays the lower of the two integers.